Lectures on Zeta Functions, L-Functions and Modular Forms with Some Physical Applications

A Window Into Zeta and Modular Physics MSRI Publications Volume 57, 2010

Lectures on zeta functions, L-functions and modular forms with some physical applications

FLOYD L. WILLIAMS

Introduction

We present nine lectures that are introductory and foundational in nature. The basic inspiration comes from the Riemann zeta function, which is the starting point. Along the way there are sprinkled some connections of the material to physics. The asymptotics of Fourier coefﬁcients of zero weight modular forms, for example, are considered in regards to black hole entropy. Thus we have some interests also connected with Einstein’s general relativity. References are listed that cover much more material, of course, than what is attempted here. Although his papers were few in number during his brief life, which was cut short by tuberculosis, Georg Friedrich Bernhard Riemann (1826–1866) ranks prominently among the most outstanding mathematicians of the nineteenth cen- tury. In particular, Riemann published only one paper on number theory [32]: “Uber¨ die Anzahl der Primzahlen unter einer gegebenen Grosse”,¨ that is, “On the number of primes less than a given magnitude”. In this short paper prepared for Riemann’s election to the Berlin Academy of Sciences, he presented a study of the distribution of primes based on complex variables methods. There the now famous Riemann zeta function

def 1 1 .s/ ; (0.1) D ns n 1 XD deﬁned for Re s > 1, appears along with its analytic continuation to the full complex plane ރ, and a proof of a functional equation (FE) that relates the values .s/ and .1 s/. The FE in fact was conjectured by Leonhard Euler, who also obtained in 1737 (over 120 years before Riemann) an Euler product

7 8 FLOYD L. WILLIAMS representation 1 .s/ s .Re s > 1/ (0.2) D 1 p pY>0 of .s/ where the product is taken over the primes p. Moreover, Riemann introduced in that seminal paper a query, now called the Riemann Hypothesis (RH), which to date has defied resolution by the best mathematical minds. Namely, as we shall see, .s/ vanishes at the values s 2n, where n 1; 2; 3;::: ; these D D are called the trivial zeros of .s/. The RH is the (yet unproved) statement that 1 if s is a zero of that is not trivial, the real part of s must have the value 2 ! Regarding Riemann’s analytic approach to the study of the distribution of primes, we mention that his main goal was to set up a framework to facilitate a proof of the prime number theorem (which was also conjectured by Gauss) which states that if .x/ is the number of primes x, for x ޒ a real number, 2 then .x/ behaves asymptotically (as x ) as x= log x. That is, one has ! 1 (precisely) that .x/ lim 1; (0.3) x x= log x D !1 which was independently proved by Jacques Hadamard and Charles de la Vallee-´ Poussin in 1896. A key role in the proof of the monumental result (0.3) is the fact that at least all nontrivial zeros of .s/ reside in the interior of the critical strip 0 Re s 1. Riemann’s deep contributions extend to the realm of physics as well - Rie- mannian geometry, for example, being the perfect vehicle for the formulation of Einstein’s gravitational field equations of general relativity. Inspired by the definition (0.1), or by the Euler product in (0.2), one can construct various other zeta functions (as is done in this volume) with a range of applications to physics. A particular zeta function that we shall consider later will bear a particular relation to a particular solution of the Einstein field equations — namely a black hole solution; see my Speaker’s Lecture. There are quite many ways nowadays to find the analytic continuation and FE of .s/. We shall basically follow Riemann’s method. For the reader’s benefit, we collect some standard background material in various appendices. Thus, to a large extent, we shall attempt to provide details and completeness of the material, although at some points (later for example, in the lecture on modular forms) the goal will be to present a general picture of results, with some (but not all) proofs. Special thanks are extended to Jennie D’Ambroise for her competent and thoughtful preparation of all my lectures presented in this volume. LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 9

CONTENTS Introduction 7 1. Analytic continuation and functional equation of the Riemann zeta function 9 2. Special values of zeta 17 3. An Euler product expansion 21 4. Modular forms: the movie 30 5. Dirichlet L-functions 46 6. Radiation density integral, free energy, and a ﬁnite-temperature zeta function 50 7. Zeta regularization, spectral zeta functions, Eisenstein series, and Casimir energy 57 8.Epsteinzetameetsgravityinextradimensions 66 9. Modular forms of nonpositive weight, the entropy of a zero weight form, and an abstract Cardy formula 70 Appendix 78 References 98

Lecture 1. Analytic continuation and functional equation of the Riemann zeta function

Since 1=ns 1=nRe s, the series in (0.1) converges absolutely for Re s > 1. j j D Moreover, by the Weierstrass M-test, for any ı > 0 one has uniform convergence of that series on the strip def S s ރ Re s > 1 ı ; ı Df 2 j C g since 1=ns 1=nRe s < 1=n1 ı on S , with j j D C ı 1 1 < : n1 ı 1 n 1 C XD def Since any compact subset of the domain S0 s ރ Re s > 1 is contained Df 2 j g in some Sı, the series, in particular, converges absolutely and uniformly on compact subsets of S0. By Weierstrass’s general theorem we can conclude that the Riemann zeta function .s/ in (0.1) is holomorphic on S0 (since the terms 1=ns are holomorphic in s) and that termwise differentiation is permitted: for Re s > 1 1 log n .s/ : (1.1) 0 D ns n 1 XD We wish to analytically continue .s/ to the full complex plane. For that purpose, we begin by considering the world’s simplest theta function .t/, deﬁned 10 FLOYDL.WILLIAMS for t > 0:

def 2 1 2 .t/ e n t 1 2 e n t (1.2) D D C n ޚ n 1 X2 XD where ޚ denotes the ring of integers. It enjoys the remarkable property that its values at t and t inverse (i.e. 1=t) are related: .1=t/ .t/ : (1.3) D pt The very simple formula (1.3), which however requires some work to prove, is called the Jacobi inversion formula. We set up a proof of it in Appendix C, based on the Poisson Summation Formula proved in Appendix C. One can of course deﬁne more complicated theta functions, even in the context of higher- dimensional spaces, and prove analogous Jacobi inversion formulas. For s ރ deﬁne 2 def .t/ 1 J.s/ 1 t s dt: (1.4) D 2 Z1 By Appendix A, J.s/ is an entire function of s, whose derivative can be obtained, in fact, by differentiation under the integral sign. One can obtain both the analytic continuation and the functional equation of .s/ by introducing the sum

def 1 I.s/ 1.n2/ se t t s 1 dt; (1.5) D n 1 Z0 XD 1 which we will see is well-deﬁned for Re s > 2 , and by computing it in different ways, based on the inversion formula (1.3). Recalling that the gamma function .s/ is given for Re s > 0 by

def .s/ 1 e t t s 1 dt (1.6) D Z0 we clearly have

def 1 1 I.s/ s .s/ s.2s/.s/; (1.7) D n2s D n 1 XD 1 so that I.s/ is well-deﬁned for Re 2s > 1: Re s > 2 . On the other hand, by the change of variables u t=n2 we transform the integral in (1.5) to obtain D

1 2 I.s/ 1 e n t t s 1 dt: D n 1 Z0 XD LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 11

We can interchange the summation and integration here by noting that

1 1 n2t s 1 1 1 n2t Re s 1 e t dt e t dt I.Re s/ < 0 D 0 D 1 n 1 Z n 1 Z XD ˇ ˇ XD 1 ˇ ˇ for Re s > 2 ; thus

1 2 .t/ 1 I.s/ 1 e n t t s 1 dt 1 t s 1 dt D D 2 Z0 n 1 Z0 XD 1 .t/ 1 .t/ 1 t s 1 dt 1 t s 1 dt; (1.8) D 2 C 2 Z0 Z1 by (1.2). Here 1 1 s 1 s 1 1 t dt lim t dt (1.9) 0 D " 0C " D s Z ! Z 1 for Re s > 0. In particular (1.9) holds for Re s > 2 , and we have 1 .t/ 1 1 1 1 t s 1 dt .t/t s 1 dt : (1.10) 2 D 2 2s Z0 Z0 By the change of variables u 1=t, coupled with the Jacobi inversion formula D (1.3), we get

1 1 1 1 1 .t/t s 1 dt t 1 s dt .t/t 2 t 1 s dt D t D Z0 Z1 Z1 1 1 1 1 ..t/ 1/ t 2 s dt t 2 s dt D C Z1 Z1 1 1 1 3 1 ..t/ 1/ t 2 s dt u 2 s .s 2 / 1 du D C C D Z1 Z0 1 1 1 ..t/ 1/ t 2 s dt ; 1 D 1 C s Z 2 1 where we have used (1.9) again for Re s > 2 . Together with equations (1.8) and (1.10), this gives

1 1 1 1 1 1 .t/ 1 I.s/ ..t/ 1/ t 2 s dt t s 1 dt 1 D 2 1 C 2.s / 2s C 1 2 Z 2 Z 1 .t/ 1 1 1 1 t s 1 t 2 s dt ; D 2 C C 2s 1 2s Z1 12 FLOYDL.WILLIAMS which with equation (1.7) gives

1 .t/ 1 1 1 1 s.2s/.s/ t s 1 t 2 s dt ; (1.11) D 2 C C 2s 1 2s Z1 1 for Re s > 2 . Finally, in (1.11) replace s by s=2, to obtain

s 1 .t/ 1 s s 1 1 1 s=2.s/ t 2 1 t 2 2 dt (1.12) 2 D 2 C C s 1 s Z1 for Re s > 1. Since z.z/ .z 1/, we have s s 2 s 1 , which D C 2 D 2 C proves: THEOREM 1.13. For Re s > 1 we can write s s s 2 .t/ 1 s s 2 2 1 2 1 2 1 .s/ t t dt : D s 2 C C s .s 1/ 2 s 1 2 Z1 2 2 C The integral 11 in this equality is an entire function of s, since, by (1.4), it equals J s 1 J s 1 . Also, since 1= .s/ is an entire function of s, 2 R C 2 it follows that the right-hand side of the equality in Theorem 1.13 provides for the analytic continuation of .s/ to the full complex plane, where it is observed that .s/ has only one singularity: s 1 is a simple pole. D The fact that 1 1=2 allows one to compute the corresponding residue: 2 D s 1 2 2 lim .s 1/.s/ lim 1: s 1 D s 1 s D 1 D ! ! 2 2 An equation that relates the values .s/ and .1 s/, called a functional equation, easily follows from the preceding discussion. In fact deﬁne

def s=2 s XR.s/ .s/ (1.14) D 2 for Re s > 1 and note that the right-hand side of equation (1.12) (which provides for the analytic continuation of XR.s/ as a meromorphic function whose simple poles are at s 0 and s 1) is unchanged if s there is replaced by 1 s: D D THEOREM 1.15 (THE FUNCTIONAL EQUATION FOR .s/). Let XR.s/ be given by (1.14) and analytically continued by the right-hand side of the (1.12). Then XR.s/ XR.1 s/ for s 0; 1. D ¤ One can write the functional equation as

s 1 2 s s 2 s .s/ C .s/ .1 s/ 2 2 (1.16) 1s 1 s D 2 1 s D 2 2 LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 13

s 1 1 s for s 0; 1, multiply the right-hand side here by 1 2 = 2 , use the ¤ 1 s 1 s 3 s D identity , and thus also write 2 2 D 2 1 s 2 s .s 1/.s/ .1 s/ C 2 ; (1.17) D 3 s 2 2 an equation that will be useful later when we compute 0.0/. For the computation of 0.0/ we make use of the following result, which is of independent interest. Œx denotes the largest integer that does not exceed x ޒ. 2 THEOREM 1.18. For Re s > 1, 1 1 Œx x 1 dx .s/ s 1 C 2 D s 1 C 2 C xs 1 Z1 C (1.19) 1 Œx x 1 s 1 dx: D s 1 C C xs 1 Z1 C dx That these two expressions for .s/ are equal follows from the equality 11 xsC1 1 for Re s > 0; this with the inequalities 0 x Œx < 1 allows one to deduce D s R that the improper integrals there converge absolutely for Re s > 0. We base the proof of Theorem 1.18 on a general observation:

LEMMA 1.20. Let .x/ be continuously differentiable on a closed interval Œa; b. Then, for c ޒ, 2 b b x c 1 .x/ dx b c 1 .b/ a c 1 .a/ .x/ dx: 2 0 D 2 2 Za Za In particular for Œa; b Œn;n 1, with n ޚ one gets D C 2 n 1 .n 1/ .n/ n 1 C x Œx 1 .x/ dx C C C .x/ dx: 2 0 D 2 Zn Zn PROOF. The ﬁrst assertion is a direct consequence of integration by parts. Using n 1 it, one obtains for the choice c n the second assertion: n C Œx0.x/ dx n 1 D D C n .x/ dx (since Œx n for n x < n 1); hence n 0 D C R R n 1 C 1 x Œx 2 0.x/ dx n Z n 1 C 1 x n 2 0.x/ dx D n Z n 1 ) 1 1 C n 1 n 2 .n 1/ n n 2 .n/ .x/ dx D C C n n 1 Z 1 .n 1/ 1.n/ C .x/ dx; ˜ D 2 C C 2 Zn 14 FLOYDL.WILLIAMS

As a ﬁrst application of the lemma, note that for integers m2; m1 with m2 > m1,

m2 Œ.n 1/ .n/ C C n m1 XD m2 m2 .n 1/ .n/ D C C n m1 n m1 XD XD .m1 1/ .m1 2/ .m2 1/ .m1/ .m1 1/ .m2/ D C C C CC C C C C CC m2

.m2 1/ .m1/ 2 .n/: D C C C n m1 1 DXC m2 n 1 m2 1 Also n m1 n C m1 C . Therefore D D P R R m2 m2 .m2 1/ .m1/ 1 C C .n/ Œ.n 1/ .n/ 2 C D 2 C C n m1 1 n m1 DXC XD m2 n 1 m2 n 1 C .x Œx 1 / .x/ dx C .x/ dx D 2 0 C n m1 n n m1 n XD Z XD Z m2 1 1 m2 1 (by Lemma 1.20), which equals C .x Œx / .x/ dx C .x/ dx. m1 2 0 C m1 Thus R R m2 .m2 1/ .m1/ .n/ C D 2 n m1 1 DXC m2 1 m2 1 1 C .x/ dx C .x Œx / .x/ dx (1.21) C C 2 0 Zm1 Zm1 for .x/ continuously differentiable on Œm1; m2 1. Now choose m1 1 and C D def .x/ x s D dx 1 s for x > 0, Re s > 1. Then 11 xs s 1 . Also .m2 1/ .m2 1/ 0 D C D C ! as m2 , since Re s > 0. Thus in (1.21) let m2 : ! 1 R ! 1

1 1 1 1 1 1 s 1 s 2 .x Œx 2 /. sx / dx: n D C s 1 C 1 n 2 Z XD That is, for Re s > 1 we have 1 1 1 1 1 .Œx x / .s/ 1 s 1 C 2 dx; D C ns D 2 C s 1 C xs 1 n 2 Z1 C XD LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 15 which proves Theorem 1.18. We turn to the second integral in equation (1.19), which we denote by

def .Œx x/ f.s/ 1 dx D xs 1 Z1 C n .Œx x/ for Re s > 0. We can write f.s/ limn 1 sC1 dx, where D !1 x n n 1 j 1 R n 1 j 1 .Œx x/ .Œx x/ j x dx C dx C dx; (1.22) s1 s1 s 1 1 x D j x D j x Z C j 1 Z C j 1 Z C XD XD since Œx j for j x < j 1. That is, f.s/ j1 1 aj .s/ where D C D D j 1 def j x j 1 1 P1 1 1 a .s/ C dx j s 1 s s s 1 s 1 D j x D s j .j 1/ s 1 j .j 1/ Z C C C for s 0; 1, and where for the second term here s 1 is a removable singularity: ¤ D 1 1 1 lim .s 1/ 0: s 1 s 1 j s 1 .j 1/s 1 D ! C Similarly, for the ﬁrst term s 0 is a removable singularity. That is, the aj .s/ D are entire functions. In particular each aj .s/ is holomorphic on the domain def D s ރ Re s > 0 . At the same time, for Re s > 0 we have C Df 2 j g WD j 1 C dx 1 1 1 aj .s/ j j x 1 D j .j 1/ Zj C C (where the inequality comes from j x x j 1 for j x j 1); moreover j jD C n 1 1 1 1 1 1 1 1 j .j 1/ D .n 1/ ) j .j 1/ D j 1 j 1 XD C C XD C (i.e. 1=.n 1/ 0 as n for > 0). Hence, by the M-test, 1 aj .s/ C ! ! 1 j 1 converges absolutely and uniformly on D (and in particular on compactD sub- C P sets of DC). f.s/ is therefore holomorphic on DC, by the Weierstrass theorem. Of course, in equation (1.19),

Œx x 1 s 1 C 2 dx sf.s/ 1 xs 1 D C 2 Z1 C is also a holomorphic function of s on DC. We have deduced: 16 FLOYDL.WILLIAMS

COROLLARY 1.23. Let 1 def .Œx / f.s/ 1 2 dx: D xs 1 Z1 C Then f.s/ is well-deﬁned for Re s > 0 and is a holomorphic function on the def domain D s ރ Re s > 0 . For Re s > 1 one has (by Theorem 1.18) C Df 2 j g 1 .s/ 1 sf.s/: (1.24) D s 1 C C From this we see that .s/ admits an analytic continuation to DC. Its only singularity there is a simple pole at s 1 with residue lims 1.s 1/.s/ 1, D ! D as before. This result is obviously weaker than Theorem 1.13. However, as a further application we show that 1 lim .s/ (1.25) s 1 s 1 D ! where def 1 1 1 lim 1 log n (1.26) D n C 2 C 3 CC n !1 is the Euler–Mascheroni constant ; 0:577215665. By the continuity (in ' particular) of f.s/ at s 1, f.1/ lims 1 f.s/. That is, by (1.24), we have D D ! 1 lim .s/ lim 1 sf.s/ s 1 s 1 D s 1 C ! ! n 1 j 1 .1:22/ j x 1 f.1/ 1 lim C dx 2 D C D C n j x !1j 1 Z n 1 XD 1 1 lim log .j 1/ log j D C n j 1 C !1j 1 C XD n 1 n 1 1 1 lim log.j 1/ log j D C n j 1 C !1j 1 C j 1 XD XD n 1 1 1 lim log n D C n j 1 !1j 1 XD C 1 1 1 1 lim 1 1 log n D C n C C 2 C 3 CC n !1 ; D as desired. LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 17

Since s 1 is a simple pole with residue 1, .s/ has a Laurent expansion D

1 1 k .s/ 0 .s 1/ (1.27) D s 1 C C k k 1 XD on a deleted neighborhood of 1. By equation (1.25), 0 . One can show D that, in fact, for k 0; 1; 2; 3;::: D k n k k 1 . 1/ .log l/ .log n/ C k lim ; (1.28) D k! n l k 1 !1 l 1 C XD a result we will not need (except for the case k 0 already proved) and thus D which we will not bother to prove. The inversion formula (1.3), which was instrumental in the approach above to the analytic continuation and FE of .s/, provides for a function F.t/, t > 0, def that is invariant under the transformation t 1=t. Namely, let F.t/ t 1=4.t/. ! D Then (1.3) is equivalent to statement that F.1=t/ F.t/, for t > 0. D Lecture 2. Special values of zeta

In 1736, L. Euler discovered the celebrated special values result n 1 2n . 1/ .2/ B2n .2n/ C (2.1) D 2.2n/! for n 1; 2; 3;::: , where Bj is the j-th Bernoulli number, defined by D z 1 B j zj ; ez 1 D j ! j 0 XD for z < 2, which is the Taylor expansion about z 0 of the holomorphic j j def D function h.z/ z=.ez 1/, which is defined to be 1 at z 0. Since ez 1 D D vanishes if and only if z 2i n, for n ޚ, the restriction z < 2 means that z D 2 j j the denominator e 1 vanishes only for z 0. The Bj were computed by D Euler up to j 30. Here are the first few values: D B0 B1 B2 B3 B4 B5 B6 B7 B8 B9 B10 B11 B12 B13 (2.2) 1 1 1 0 1 0 1 0 1 0 5 0 691 0 2 6 30 42 30 66 2730 def In general, B >1 0: To see this let H.z/ h.z/ z=2 for z < 2, which odd D D C j j we claim is an even function. Namely, for z 0 the sum z=.ez 1/ z=.e z 1/ ¤ C equals z by simplification: D z z z z H. z/ z H.z/: D e z 1 2 D ez 1 C 2 D 18 FLOYDL.WILLIAMS

Then z 1 B2j 2j 1 B2j 1 2j 1 B0 B1z z C z 2 C C C .2j /! C .2j 1/! C j 1 j 1 XD XD C z 1 B j zj H.z/ H. z/ D 2 C j ! D D j 0 XD z 1 B2j 2j 1 B2j 1 2j 1 B0 B1. z/ . z/ C . z/ ; D 2 C C C .2j /! C .2j 1/! C j 1 j 1 XD XD C which implies

1 B2j 1 2j 1 0 .1 2B1/z 2 C z ; D C C .2j 1/! C j 1 XD C 1 and consequently B1 2 and B2j 1 0 for j 1, as claimed. By formula D C D (2.1) (in particular)

2 4 6 1 1 1 1 1 1 .2/ ; .4/ ; .6/ ; (2.3) D n2 D 6 D n4 D 90 D n6 D 945 n 1 n 1 n 1 XD XD XD 2 2 the ﬁrst formula, 1 1=n =6, being well-known apart from knowledge n 1 D of the zeta function .Ds/. We provide a proof of (2.1) based on the summation P formula 1 1 1 coth a (2.4) n2 a2 D 2a 2a2 n 1 XD C for a > 0; see Appendix E on page 92. Before doing so, however, we note some other special values of zeta. As we have noted, 1= .s/ is an entire function of s. It has zeros at the points s 0; 1; 2; 3; 4;::: . By Theorem 1.13 and the remarks that follow its D statement we therefore see that for n 1; 2; 3; 4;::: , D n 1 1 . 2n/ 0; .0/ : (2.5) D 2. n 1/ D D 2.1/ D 2 C Thus, as mentioned in the Introduction, .s/ vanishes at the real points s D 2; 4; 6; 8;:::, called the trivial zeros of .s/. The value .0/ is nonzero — it equals 1 by (2.5). Later we shall check that 2 .0/ .0/ log 2 1 log 2: (2.6) 0 D D 2 LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 19 t Turning to the proof of (2.1), we take 0 < t < 2 and choose a in (2.4), D 2 obtaining successively

2 2 t 2 1 1 coth 42 ; t 2 t 2 D t 2 42n2 n 1 XD C 1 t 1 1 1 coth 2t ; 2 2 t D t 2 42n2 n 1 XD C 1 1 2 et 1 e t=2 e t=2 et=2 C C et 1 C 2 D 2.et 1/ e t=2 D 2.et=2 e t=2/ 1 cosh.t=2/ 1 t 1 1 1 coth 2t ; D 2 sinh.t=2/ D 2 2 D t C t 2 42n2 n 1 XD C t t 1 1 1 2t 2 : (2.7) et 1 C 2 D C t 2 42n2 n 1 XD C 1 Since B0 1 and B1 2 (see (2.2)), and since B2k 1 0 for k 1, we can D D C D write t def 1 B t 1 B k t k 1 2k t 2k ; et 1 D k! D 2 C .2k/! k 0 k 1 XD XD and (2.7) becomes

1 B 1 1 2k t 2k 2t 2 : (2.8) .2k/! D t 2 42n2 k 1 n 1 XD XD C For 0 < t < 2, we can use the convergent geometric series

2 k 2 2 1 t 1 4 n 2 ; (2.9) 42n2 D 1 t D t 2 42n2 k 0 42n2 XD C C to rewrite (2.8) as 2 k 1 B 1 1 1 t 2k t 2k 2t 2 (2.10) .2k/! D 42n2 42n2 k 1 n 1 k 0 XD XD XD 2 k 1 1 1 1 t 2t 2 : D 42n2 42n2 n 1 k 1 XD XD The point is to commute the summations on n and k in this equation. Now

2 k 1 2.k 1/ 2.k 1/ 1 1 1 t 1 t 1 1 1 t 1 1 42n2 42n2 D .42/k n2k .42/k n2 k 1 n 1ˇ ˇ k 1 n 1 n 1 n 1 XD XD ˇ ˇ XD XD XD XD ˇ ˇ ˇ ˇ 20 FLOYDL.WILLIAMS

2 2.k 1/ 2 k which is finite since 1 1=n .2/ < and 1 t =.4 / < , n 1 D 1 n 1 1 by the ratio test (againD for 0 < t < 2). CommutationD of the summation is P P therefore justified: 2k 2 k 1 k 1 1 B t 1 . t / 1 1 1 2. 1/ 2k 2t 2 .2k/t 2k .2k/! D 42.42/k 1 n2k D .42/k k 1 k 1 n 1 k 1 XD XD XD XD on .0; 2/. By equating coefficients, we obtain B 2. 1/k 1.2k/ 2k for k 1; .2k/! D .42/k which proves Euler’s formula (2.1). Next we turn to a proof of equation (2.6). We start with an easy consequence of the quotient and product rules for differentiation. LEMMA 2.11 (LOGARITHMIC DIFFERENTIATION WITHOUT LOGS). If .s/ .s/ .s/ F.s/ 1 2 3 ; D 4.s/ on some neighborhood of s0 ރ, where the j .s/ are nonvanishing holomorphic 2 functions there, then

F .s / .s0/ .s0/ .s0/ .s0/ 0 0 10 20 30 40 : F.s0/ D 1.s0/ C 2.s0/ C 3.s0/ 4.s0/ def 1 s def s 3 s Now choose 1.s/ 2 , 2.s/ , 4.s/ 2 , say on a small D D 2 D 2 neighborhood of s 1. For the choice of 3.s/, we write .s/ g.s/=.s 1/ D D on a neighborhood N of s 1, for s 1, where g.s/ is holomorphic on N and D ¤ g.1/ 1. This can be done since s 1 is a simple pole of .s/ with residue D D def 1; for example, see equation (1.27). Assume 0 N and take 3.s/ g.s/ on D … D N . By equation (1.17), .1 s/ 1.s/2.s/3.s/=4.s/ near s 1, so that D def D by Lemma 2.11 and introducing the function .s/ .s/=.s/, we obtain D 0 .1 s/ 0 .1 s/ s 1 1 ˇ D 2 s ˇ . ˇlog / s 1 g0.s/ 3 s 1 1ˇ : (2.12) D 2 s C 2 2 C g.s/ 2 2 ˇs 1 ˇs 1 ˇs 1 ˇs 1 ˇ D ˇ D ˇ D ˇ D ˇ ˇ ˇ ˇ If is the Euler–Mascheroniˇ ˇ constant ofˇ (1.26), the facts .ˇ1/ and D 1 2 log 2 are known to prevail, which reduces equation (2.12) to 2 D

.0/ .0/ log log 2 g .1/ 0 D C 2 C 0 C 2 1 log log 2 g .1/ ; D 2 C C 0 LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 21 since g.1/ 1 and .0/ 1 ; see (2.5). But g .1/ , as we will see in a D D 2 0 D minute; hence we have reached the conclusion that .0/ 1 log 2, which is 0 D 2 (2.6). There remains to check that g .1/ . We have 0 D def g.s/ 1 g0.1/ lim ; D s 1 s 1 ! 1 again since g.1/ 1; this in turn equals lims 1 .s/ , by equation D ! s 1 D (1.25). To obtain further special values of zeta we appeal to the special values formula . 1/np 22nn! 1 n (2.13) 2 D .2n/! for the gamma function, where n 1; 2; 3; 4;::: . This we couple with (2.1) D and the functional equation (1.16) to show that 1 B . 1/ and .1 2n/ 2n for n 1; 2; 3; 4;:::: (2.14) D 12 D 2n D 1 Namely, .1 2n/ 2n 2 .n/.2n/= 1 n , by (1.16); this in turn equals D C 2 1 2n 2 .n 1/!.2n/.2n /! C ; . 1/np22nn! by (2.13); whence (2.1) gives n 2n . 1/ .2/ .2n/.2n/! B2n .1 2n/ : D n D 2n B 1 Taking n 1 gives . 1/ 2 , by (2.2), which conﬁrms (2.14). D D 2 12 Lecture 3. An Euler product expansion For a function f.n/ deﬁned on the set ޚ 1; 2; 3;::: of positive integers C Df g one has a corresponding zeta function or Dirichlet series

def 1 f.n/ .s/ ; f D ns n 1 XD defined generically for Re s sufficiently large. If f.n/ 1 for all n ޚ , for D 2 C example, then for Re s > 1, f .s/ is of course just the Riemann zeta function .s/, which according to equation (0.2) of the Introduction has an Euler product 1 expansion .s/ s over the primes P in ޚ . It is natural to inquire D p P 1 p C whether, more generally,2 there are conditions that permit an analogous Euler Q product expansion of a given Dirichlet series f .s/. Very pleasantly, there is 22 FLOYDL.WILLIAMS an affirmative result when, for example, the f.n/ are Fourier coefficients (see Theorem 4.32, where the n-th Fourier coefficient there is denoted by an) of certain types of modular forms, due to a beautiful theory of E. Hecke. Also see equations (3.20), (3.21) below. Rather than delving directly into that theory at this point we shall instead set up an abstract condition for a product expansion. The goal is to show that under suitable conditions on f.n/ of course the desired expansion assumes the form

def 1 f.n/ f.1/ f .s/ (3.1) D ns D 1 ˛.p/p 2s f.p/p s n 1 p P XD 2 C for some function ˛.p/ on P; seeQ Theorem 3.17 below. Here we would want to have, in particular, that f.1/ 0. Before proceeding toward a precise statement ¤ and proof of equation (3.1), we note that (again) if f.n/ 1 for all n ޚ , for D 2 C example, then for the choice ˛.p/ 0 for all p P, equation (3.1) reduces to D 2 the classical Euler product expansion of equation (0.2). Given f ޚ ޒ or ރ, and ˛ P ޒ or ރ, we assume the following W C ! W ! abstract multiplicative condition: f.np/ if p - n, f.n/f.p/ (3.2) D f.np/ ˛.p/f n if p n, C p j for .n; p/ ޚ P; here p n means that p divides n and p - n means the 2 C j opposite. Given condition (3.2) we observe first that if f.1/ 0 then f vanishes D identically, the proof being as follows. For a prime p P, (3.2) requires that 2 f.1/f.p/ f.p/, since p - 1; that is, f.p/ 0. If n ޚ with n 2, there D D 2 C exists p P such that p n, say ap n, a ޚ . Proceed inductively. If p - a, 2 j D 2 C f.a/f.p/ f.ap/ f.n/, by (3.2), so f.n/ 0, as f.p/ 0. If p a, we have D D D D j 0 f.a/f.p/ (again as f.p/ 0), and this equals f.ap/ ˛.p/f.a=p/ D D C D f.n/ ˛.p/f.a=p/, where 1 < p a (so 1 a=p < a n=p < n/. Thus C D f.a=p/ 0, by induction, so f.n/ 0, which completes the induction. Thus D D we see that if f 0 then f.1/ 0. 6 ¤ As in Appendix D (page 88) we set, m; n ޚ , 2 C def 1 if m n, d.m; n/ j D 0 if m - n. Fix a finite set of distinct primes S p1; p2;:::; p P and define g.n/ Df l g D gS .n/ on ޚC by l g.n/ f.n/ 1 d.pj ; n/ : (3.3) D j 1 YD LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 23

Fix p P p1; p2;:::; p . Then the next observation is that, for n ޚ , 2 f l g 2 C g.np/ if p - n, g.n/f.p/ n ; (3.4) D g.np/ ˛.p/g if p n ( C p j which compares with equation (3.2).

PROOF. If pj n then of course pj pn. If pj - n then pj - pn; for if pj pn then j j j p n since pj ; p are relatively prime, given that p each pj . Thus j ¤

d.pj ; n/ d.pj ; pn/ for n ޚ , 1 j l. (3.5) D 2 C

Similarly suppose p n, say bp n, with b ޚ . If pj - n=p then pj - n; for j D 2 C otherwise pj n bp again with pj ; p relatively prime, implying that pj b j D j D n=p. Thus we similarly have

d.pj ; n=p/ d.pj ; n/ for n ޚ , 1 j l, such that p n. (3.6) D 2 C j Now if n ޚ is such that p - n, then 2 C .3:3/ l .3:2/ l g.n/f.p/ f.n/f.p/ 1 d.pj ; n/ f.np/ 1 d.pj ; pn/ D j 1 .3D:5/ j 1 D D .3:3/ Q Q g.np /: D On the other hand, if p n, then j .3:3/ l g.n/f.p/ f.n/f.p/ 1 d.pj ; n/ D j 1 D Q l .3:2/ n f.np/ ˛.p/f. p / 1 d.pj ; n/ D C j 1 D l Q l .3:5/ n n f.np/ 1 d.pj ; pn/ ˛.p/f. p / 1 d.pj ; p / .3D:6/ j 1 C j 1 D D .3:3/ Q Q g.np/ ˛.p/g n ; D C p which proves (3.4). ˜

s Let h.s/ n1 1 h.n/=n be a Dirichlet series that converges absolutely, say D D at some ﬁxed point s0 ރ. Fix p P and some complex number .p/ corre- P 2 2 sponding to p such that h.np/ if p - n, h.n/.p/ (3.7) D h.np/ ˛.p/h n if p n, C p j 24 FLOYDL.WILLIAMS for n ޚ . Then 2 C

2s0 s0 1 1 d.p; n/ h.n/ h.s0/ 1 ˛.p/p .p/p : (3.8) C D ns0 n 1 XD PROOF. Deﬁne n ˛.p/h p def if p n, an s0 D 8 .pn/ j < 0 if p - n, for n ޚ . Since p pn, we have 2 C j : pn ˛.p/h h.n/ a p ˛.p/p 2s0 ; pn s0 s0 D .ppn/ D n which shows that n1 1 apn converges. The Scholium of Appendix D (page D 91) then implies that the series n1 1 d.p; n/an converges, and one has P D 1 P 1 apn d.p; n/an; (3.9) D n 1 n 1 XD XD 2s0 where the left-hand side here is ˛.p/p .s0/. Since both d.p; n/; an 0 h D if p - n, d.p; n/an an, which is also clear if p n. On the other hand, if p n, D j j def n s0 s0 an ˛.p/h .pn/ h.n/.p/ h.np/ .pn/ (3.10) D p D by equation (3.7). Equation (3.10) also holds by (3.7) in case p - n, for then both sides are zero. That is, (3.10) holds for all n 1 and equation (3.9) reduces to the statement

2s0 1 s0 ˛.p/p .s0/ Œh.n/.p/ h.np/.pn/ : (3.11) h D n 1 XD def s0 We apply the Scholium a second time, where this time we define an h.n/=n ; D since d.p; n/an an , the sum n1 1 d.p; n/an converges. By the Scholium, j j j j D n1 1 apn converges and n1 1 apn n1 1 d.p; n/an; that is, D D P D D P 1 P s0 P1 s0 h.pn/.pn/ d.p; n/h.n/n ; n 1 D n 1 D D P P 2s0 s0 which one plugs into (3.11), to obtain ˛.p/p h.s0/ .p/p h.s0/ s0 D n1 1 d.p; n/h.n/n . This proves equation (3.8). TheD proof of the main result does involve various moving parts, and it is a bit P lengthy as we have chosen to supply full details. We see, however, that the proof is elementary. One further basic ingredient is needed. Again let p1;:::; p by f l g a fixed, finite set of distinct primes in P. With f;˛ subject to the multiplicative LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 25 condition (3.2), we assume that f .s0/ converges absolutely where s0 ރ is l 2 some fixed number. For n 1, we have 0 j 1 1 d.pj ; n/ 1; therefore l s0D the series 1 .1 d.pj ; n// f.n/=n converges absolutely. We now n 1 j 1 Q show by inductionD onDl that P Q l 2s0 s0 1 ˛.pj /pj f.pj /pj f .s0/ j 1 C D l f.n/ Q 1 1 d.p ; n/ : (3.12) j s0 D n 1 j 1 n PD QD For l 1, the claim follows by (3.8) with p p1, h.n/ f.n/, .p/ f.p/. D D D D Proceeding inductively, we consider a set p1; p2;:::; pl ; pl 1 of l 1 distinct f C g C primes in P. Then l 1 C 2s0 s0 1 ˛.pj /pj f.pj /pj f .s0/ j 1 C D Q l 2s0 s0 2s0 s0 1 ˛.pl 1/pl 1 f.pl 1/pl 1 1 ˛.pj /pj f.pj /pj f .s0/ D C C C C C j 1 C D Q l 2s0 s0 1 f.n/ 1 ˛.p /p f.p /p 1 d.p ; n/ l 1 l 1 l 1 l 1 j s0 D C C C C C n 1 j 1 n PD QD 2s0 s0 1 g.n/ 1 ˛.p /p f.p /p ; (3.13) l 1 l 1 l 1 l 1 s0 D C C C C C n 1 n D where the second equality follows by inductionP and the last one by definition l s0 (3.3). We noted, just above (3.12), that n1 1 j 1.1 d.pj ; n// f.n/=n Ds0 D converges absolutely; that is, n1 1 g.n/=n converges absolutely. Thus we D P Q choose h.n/ g.n/, p pl 1, .p/ f.p/. Condition (3.7) is then a conse- D D C P D quence of equation (3.4), since pl 1 P p1; p2;:::; pl , and one is therefore C 2 f g able to apply formula (3.8) again:

2s0 s0 1 ˛.pl 1/pl 1 f.pl 1/pl 1 g.s0/ C C C C C 1 1 d.pl 1; n/ g.n/ C s0 D n 1 n D P l .3:3/ 1 1 d.pl 1; n/ C 1 d.p ; n/ f.n/ s0 j D n 1 n j 1 PD QD l 1 f.n/ 1 C 1 d.p ; n/ (3.14) j s0 D n 1j 1 n PD QD which, together with equation (3.13), allows one to complete the induction, and thus the proof of the claim (3.12). ˜ 26 FLOYDL.WILLIAMS

Now let p be the l-th positive prime, so p1 2, p2 3, p3 5, p4 7,.... l D D D D We can write the right-hand side of (3.12) as

l f.n/ f.1/ 1 1 d.p ; n/ ; j s0 C n 2 j 1 n PD QD since no pj divides 1. We show that if 2 n l then l 1 d.pj ; n/ 0: (3.15) j 1 D QD Namely, for n 2 choose q P such that q n. If no pj divides n, 1 j l, 2 j then q p1;:::; pl (since q n); hence q pl 1 (since pl is the l-th prime) and ¤ j C so q l 1. But this is impossible since n l (by hypothesis) and q n (since C q n). This contradiction proves that some pj divides n, that is, 1 d.pj ; n/, j D which gives (3.15). It follows that l f.n/ l f.n/ 1 1 d.p ; n/ f.1/ 1 1 d.p ; n/ ; j s0 j s0 n 1 j 1 n D n l 1 j 1 n PD QD DPC QD l where (using again that 0 j 1 1 d.pj ; n/ 1) we have D l f.Qn/ f.n/ f.n/ l f.n/ 1 1 d.p ; n/ 1 1 : j s0 s0 s0 s0 ˇn l 1 j 1 n ˇ n l 1 n D n 1 n n 1 n ˇ D C D ˇ D C ˇ ˇ D ˇ ˇ D ˇ ˇ ˇ P Q ˇ P ˇ ˇ P ˇ ˇ P ˇ ˇ ˇBut this difference tends to 0 as lˇ . Thatˇ is, byˇ equationˇ (3.12),ˇ theˇ limitˇ ˇ ˇ! 1 ˇ ˇ ˇ ˇ ˇ ˇ 2s0 s0 1 ˛.p/p f.p/p f .s0/ p P C 2 l Q def 2s0 s0 lim 1 ˛.pj /pj f.pj /p f .s0/ (3.16) D l j 1 C !1 QD exists, where pj the j-th positive prime, and it equals f.1/. D We have therefore ﬁnally reached the main theorem. def THEOREM 3.17. (As before, ޚ 1; 2; 3;::: and P denotes the set of pos- C Df g itive primes.) Let f ޚ ޒ or ރ and ˛ P ޒ or ރ be functions where W ! W ! f is not identically zero and where f is subject to the multiplicative condition (3.2). Then f.1/ 0. Let D ރ be some subset on which the corresponding ¤ s Dirichlet series f .s/ n1 1 f.n/=n converges absolutely. Then on D, D D 1 P˛.p/p 2s f.p/p s .s/ f.1/ C f D p P Y2 LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 27

(see equation (3.16)). In particular both 1 ˛.p/p 2s f.p/p s and p P C .s/ are nonzero on D and 2 f Q f.1/ .s/ (3.18) f 2s s D p P 1 ˛.p/p f.p/p 2 C on D (which is equation (3.1)).Q We remark (again) that the proof of Theorem 3.17 is entirely elementary, if a bit long-winded; it only requires a few basic facts about primes, and a weak version of the fundamental theorem of arithmetic — namely that an integer n 2 is divisible by a prime. As a simple example of Theorem 3.17, suppose f ޚ ޒ or ރ is not W C ! identically zero, and is completely multiplicative: f.nm/ f.n/f.m/ for all D s n; m ޚ . Assume also that s ރ is such that 1 f.n/=n converges 2 C 2 n 1 absolutely. Then D P 1 f.n/ 1 : (3.19) ns D f.p/ n 1 1 s XD p P p Q2 To see this, ﬁrst we note directly that f.1/ 0. In fact since f 0, choose ¤ 6 n ޚ such that f.n/ 0. Then f.n/ f.n 1/ f.n/f.1/, so f.1/ 1. 2 C ¤ D D D Also f satisﬁes condition (3.2) for the choice ˛ 0 (that is, ˛.p/ 0 for all D D p P). Equation (3.19) therefore follows by (3.18). In Lecture 5, we apply 2 (3.19) to Dirichlet L-functions. Before concluding this lecture, we feel some obligation to explain the pivotal, abstract multiplicative condition (3.2). This will involve, however, some facts regarding modular forms that will be discussed in the next lecture, Lecture 4. Thus suppose that f.z/ is a holomorphic modular form of weight k 4; 6; 8; 10;::: , D with Fourier expansion

1 2inz f.z/ ane (3.20) D n 0 XD on the upper half-plane C. Then there is naturally attached to f.z/ a Dirichlet series 1 a .s/ n ; (3.21) f D ns n 1 XD called a Hecke L-function, which is known to converge absolutely for Re s > k, and which is holomorphic on this domain. Actually, if f.z/ is a cusp form (i.e., k a0 0) then .s/ is holomorphic on the domain Re s > 1 . As in the D f C 2 case of the Riemann zeta function, Hecke theory provides for the meromorphic 28 FLOYDL.WILLIAMS continuation of f .s/ to the full complex plane, and for an appropriate functional equation for f .s/. For each positive integer n 1; 2; 3;::: , there is an operator T .n/ (called a D Hecke operator) on the space of modular forms of weight k given by nz da .T .n/f /.z/ nk 1 f C d k : (3.22) D d 2 d>0 a ޚ=dޚ dXn 2X j Here the inner sum is over a complete set of representatives a in ޚ for the cosets ޚ=dޚ. We shall be interested in the case when f.z/ is an eigenfunction of all Hecke operators: T .n/f .n/f for all n 1, where f 0 and .n/ ރ. We D ¤ 2 assume also that a1 1, in which case f.z/ is called a normalized simultaneous D eigenform. For such an eigenform it is known from the theory of Hecke operators that the Fourier coefﬁcients and eigenvalues coincide for n 1 an .n/ W D for n 1. Moreover the Fourier coefﬁcients satisfy the “multiplicative” condition k 1 an1 an2 d a 2 (3.23) D n1n2=d d>0 d nX1;d n2 j j for n1; n2 1. In particular for a prime p P and an integer n 1, condition 2 (3.23) clearly reduces to the simpler condition

anp if p - n, anap k 1 (3.24) D anp p a 2 if p n, C np=p j which is the origin of condition (3.2), where we see that in the present context k 1 we have f.n/ an and ˛.p/ p ; f.n/ here is the function f ޚ ރ D D W ! of condition (3.2), of course, and is not the eigenform f.z/. By Theorem 3.17, therefore, the following strong result is obtained.

THEOREM 3.25 (EULER PRODUCT FOR HECKE L-FUNCTIONS). Let f.z/ be a normalized simultaneous eigenform of weight k (as desribed above), and let f .s/ be its corresponding Hecke L-function given by definition (3.21) for Re s > k. Then, for Re s > k, 1 .s/ ; (3.26) f k 1 2s s D p P 1 p app 2 C where ap is the p-th FourierQ coefficient f.z/; see equation(3.20). If , moreover, f.z/ is a cusp form (i.e., the 0-th Fourier coefficient a0 of f.z/ vanishes), then for Re s > 1 k , .s/ converges (in fact absolutely) and formula (3.26) holds. C 2 f The following example is important, though no proofs (which are quite involved) LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 29 are supplied. If

def 1 .z/ eiz=12 .1 e2inz/ (3.27) D n 1 YD is the Dedekind eta function on C, then the Ramanujan tau function .n/ on ޚC is deﬁned by the Fourier expansion

1 .z/24 .n/e2inz; (3.28) D n 1 XD which is an example of equation (3.20), where in fact .z/24 is a normalized simultaneous eigenform of weight k 12. It turns out, remarkably, that every D .n/ is real and is in fact an integer. For example, .1/ 1, .2/ 24, D D .3/ 252, .4/ 1472, .5/ 4830. Note also that, since the sum in (3.27) D D D starts at n 1, .z/24 is a cusp form. By Theorem 3.25 we get: D k COROLLARY 3.29. For Re s > 1 7, C 2 D 1 .n/ 1 : (3.30) ns D 1 p11 2s .p/p s n 1 p P XD 2 C The Euler product formula (3.30)Q was actually proved ﬁrst by L. Mordell (in 1917, before E. Hecke) although it was claimed earlier to be true by S. Ra- manujan. Since we have introduced the Dedekind eta function .z/ in (3.27), we check, as a ﬁnal point, that it is indeed holomorphic on C. For

def 2inz def 1 an.z/ e and G.z/ .1 an.z//; D D n 1 C QD iz=12 write .z/ e G.z/. The product G.z/ converges absolutely on C since D 2ny ޒ n1 1 an.z/ n1 1 e (for z x iy, x; y ; y > 0) is a convergent D j jD D 2y D C 2 geometric series as e < 1. We note also that an.z/ 1 since (again) P 2nyP ¤ an.z/ e < 1 for n 1. If K is any compact subset, then the j jD C continuous function Im z on K has a positive lower bound B: Im z B > 0 for every z K. Hence 2 2n Im z 2nB an.z/ e e on K; j jD 2nB 2B where n1 1 e is a convergent geometric series as e < 1 for B > 0. D Therefore the series 1 an.z/ converges uniformly on compact subsets of P n 1 (by the M-test), whichD means that the product G.z/ converges uniformly on C P compact subsets of C. That is, G.z/ is holomorphic on C (as the an.z/ are holomorphic on C), and therefore .z/ is holomorphic on C. 30 FLOYDL.WILLIAMS

Lecture 4. Modular forms: the movie

In the previous lecture we proved an Euler product formula for Hecke L- functions, in Theorem 3.25 which followed as a concrete application of Theo- rem 3.17. That involved, in part, some notions/results deferred to the present lecture for further discussion. Here the attempt is to provide a brief, kaleidoscopic tour of the modular universe, whose space is Lobatchevsky–Poincare´ hyperbolic space, the upper half-plane C, and whose galaxies of stars are modular forms. As no universe would be complete without zeta functions, Hecke L-functions play that role. In particular we gain, in transit, an enhanced appreciation of Theorem 3.25. There are many fine texts and expositions on modular forms. These obviously venture much further than our modest attempt here which is designed to serve more or less as a limited introduction and reader’s guide. We recommend, for example, the books of Audrey Terras [35], portions of chapter three, (also note her lectures in this volume) and Tom Apostol [2], as supplements. We begin the story by considering a holomorphic function f.z/ on C that satisfies the periodicity condition f.z 1/ f.z/. By the remarks following C D Theorem B.7 of the Appendix (page 84), f.z/ admits a Fourier expansion (or q-expansion) n 2inz f.z/ anq.z/ ane (4.1) D D n ޚ n ޚ X2 X2 def 2iz on , where q.z/ e , and where the an are given by formula (B.6). We C D say that f.z/ is holomorphic at infinity if an 0 for every n 1: D

1 2inz f.z/ ane (4.2) D n 0 XD on . Let G SL.2; ޒ/ denote the group of 2 2 real matrices g a b with C D D c d determinant 1, and let SL.2; ޚ/ G denote the subgroup of elements D D a b with a; b; c; d ޚ. The standard linear fractional action of G on , D c d 2 C given by def az b g z C for .g; z/ G (4.3) D cz d 2 C 2 C C restricts to any subgroup of G, and in particular it restricts to . A (holomorphic) modular form of weight k ޚ, k 0, with respect to , is 2 a holomorphic function f.z/ on C that satisﬁes the following two conditions: (M1) f. z/ .cz d/kf.z/ for a b , z . D C D c d 2 2 C (M2) f.z/ is holomorphic at inﬁnity. LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 31

def Here we note that for the case of T 1 1 , we have z z 1 D D 0 1 2 D C by (4.3). Then f.z 1/ f.z/ by (M1), which means that condition (M2) C D is well-defined, and therefore f.z/ satisfies equation (4.2), which justifies the statement of equation (3.20) of Lecture 3. One can also consider weak modular forms, where the assumption that an 0 for every n 1 is relaxed to allow D finitely many negative Fourier coefficients to be nonzero. One can consider, moreover, modular forms with respect to various subgroups of . There are def 1 0 two other quick notes to make. First, if 1 , then by (4.3) D D 0 1 2 and (M1) we must have f.z/ . 1/kf.z/ which means that f.z/ 0 if k is D odd. For this reason we always assume that k is even. Secondly, condition (M1) is equivalent to the following two conditions: (M1) f.z 1/ f.z/, and 0 C D (M1) f. 1=z/ zk f.z/ for z . 00 D 2 C For we have already noted that (M1) (M1)0, by the choice T . Also def 0 1 ) D choose S . Then by (4.3) and (M1), condition (M1) follows. D D 1 0 2 00 Conversely, the conditions (M1) and (M1) together, for k 0 even, imply 0 00 condition (M1) since the two elements T; S generate ; a proof of this is 2 provided in Appendix F (page 96). Basic examples of modular forms are provided by the holomorphic Eisenstein series Gk.z/, which serve in fact as building blocks for other modular forms:

def 1 Gk.z/ (4.4) D .m nz/k .m;n/ ޚ ޚ .0;0/ 2 X f g C for z , k 4; 6; 8; 10; 12;::: . The issue of absolute or uniform convergence 2 C D of these series rests mainly on the next observation, whose proof goes back to Chris Henley [2]. Given A; ı > 0 let

def S .x; y/ ޒ2 x A; y ı (4.5) A;ı Df 2 j j j g be the region , as illustrated: C y

ı x 0 A A 32 FLOYDL.WILLIAMS

LEMMA 4.6. There is a constant K K.A;ı/> 0, depending only on A and D ı, such that for any .x; y/ S and .a; b/ ޒ2 with b 0 the inequality 2 A;ı 2 ¤ .a bx/2 b2y2 C C K (4.7) a2 b2 C holds. In fact one can take 2 def ı K : (4.8) D 1 .A ı/2 C C def PROOF. Given .x; y/ S and .a; b/ ޒ2 with b 0, let q a=b. Then 2 A;ı 2 ¤ D (4.7) amounts to .q x/2 y2 C C K: 1 q2 C Two cases are considered. First, if q A ı, then 1 q2 1 .A ı/2, so j j C C C C 1 1 : 1 q2 1 .A ı/2 C C C Also .q x/2 y2 y2 ı2 (since y ı for .x; y/ S ). Therefore C C 2 A;ı .q x/2 y2 ı2 C C K; 1 q2 1 .A ı/2 D C C C with K as in (4.8). If instead q > A ı, we have 1= q < 1=.A ı/, so x =q x =.A ı/. j j C j j C j j j j C Use the triangular inequality and the fact that x A for .x; y/ S to write j j 2 A;ı x x x A ı 1 1 1 j j 1 : C q q A ı A ı D A ı ˇ ˇ ˇ ˇ C C C ˇ ˇ ˇ ˇ 2 2 ˇ ˇ ı ˇ ˇ 2 q ı That is, q ˇ x ˇ q ˇ , whichˇ implies .q x/ , or again j C j j jA ı C .A ı/2 C C .q x/2 y2 .q x/2 ı2 q2 C C C : (4.9) 1 q2 1 q2 .A ı/2 1 q2 C C C C def On the other hand, f.x/ x2=.1 x2/ is a strictly increasing function D C on .0; / since f .x/ 2x=..1 x2/2/ is positive for x > 0. Thus, since 1 0 D C q > A ı, we have j j C q2 .A ı/2 f. q />f.A ı/ C ; 1 q2 D j j C D 1 .A ı/2 C C C which leads to .q x/2 y2 ı2 .A ı/2 C C C 1 q2 .A ı/2 1 .A ı/2 C C C C LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 33 by the second inequality in (4.9). But the right-hand side is again the constant K of (4.8). This concludes the proof. ˜ 2 2 2 2 .a bx/ b y a 1 Now suppose b 0, but a 0. Then C C 1 > ; which D ¤ a2 b2 D a2 D 2 by Lemma 4.6 says that C

2 2 2 .a bx/ b y def 1 C C K1 min ; K a2 b2 D 2 C for .x; y/ SA;ı, .a; b/ ޒ ޒ .0; 0/ . Hence if z SA;ı, say z x iy, and 2 2 f 2g 2 2 2 D2 C2 .m; n/ ޚ ޚ .0; 0/ , we get m nz .m nx/ n y .m n /K1 2 2 f g j C j D C ˛C ˛=2 C ˛ D K1 m ni . This implies, for ˛ 0, that m nz K m ni , or j C j j C j 1 j C j 1 1 : (4.10) m nz ˛ K˛=2 m ni ˛ j C j 1 j C j def Moreover — setting for convenience ޚ2 ޚ ޚ .0; 0/ — we know from D ˛f g results in Appendix G that ޚ2 1= m ni converges for ˛ > 2. This .m;n/ j C j shows that G .z/ converges absolutely2 and uniformly on every S for k > 2, k P A;ı which (since k is even) is why we take k 4; 6; 8; 10; 12;::: in (4.4). In D particular, since any compact subset of C is contained in some SA;ı, the holomorphicity of Gk.z/ on C is established. Since the map .m; n/ .m n; n/ is a bijection of ޚ2, we have ‘ 1 1 G .z 1/ G .z/: k k k k C D 2 .m n nz/ D 2 .m n n nz/ D .m;n/ ޚ .m;n/ ޚ P2 C C P2 C C Similarly,

1 1 1 1 G zk zk ; k z k k k D 2 .m n=z/ D 2 .mz n/ D 2 .nz m/ .m;n/ ޚ .m;n/ ޚ .m;n/ ޚ C X2 X2 X2 2 since the map .m; n/ .n; m/ is a bijection of ޚ . This shows that Gk.z/ ‘ satisﬁes the conditions (M1)0 and (M1)00. To complete the argument that the G .z/, for k even 4, are modular forms k of weight k, we must check condition (M2). Although this could be done more directly, we take the route whereby the Fourier coefﬁcients of Gk.z/ are actually computed explicitly. For this, consider the function

def 1 k .z/ (4.11) D .z m/k m ޚ X2 C 34 FLOYDL.WILLIAMS on for k ޚ, k 2. The inequality (4.10) gives C 2 1 1 1 (4.12) m z k Kk=2 m i k D Kk=2.m2 1/k=2 j C j 1 j C j 1 C for z S . Since 2 A;ı

1 1 1 1 1 1 2 1 2 < .m2 1/k=2 D C .m2 1/k=2 C m2.k=2/ mk 1 m ޚ m 1 m 1 X2 C XD C XD D for k > 1, we see that k .z/ converges absolutely and uniformly on every SA;ı, and is therefore a holomorphic function on C such that 1 1 k.z 1/ k .z/: C D .z m 1/k D .z m/k D m ޚ m ޚ X2 C C X2 C Thus (again) there is a Fourier expansion

2inz .z/ an.k/e (4.13) k D n ޚ X2 on , where by formula (B.6) of page 84 (with the choice b1 0, b2 ) C D D 1 1 2in.t ib/ an.k/ .t ib/e dt (4.14) D k C C Z0 for n ޚ, b > 0. 2 PROPOSITION 4.15. In the Fourier expansion (4.13), an.k/ 0 for n 0 and k k 1 D an.k/ . 2i/ n =.k 1/! for n 1. Therefore D k . 2i/ 1 .z/ nk 1e2inz k D .k 1/! n 1 XD is the Fourier expansion of the function .z/ of (4.11) on . Here k ޚ, k C 2 k 2 as in (4.11). def 2int k PROOF. For ﬁxed n ޚ and b > 0, deﬁne hm.t/ e =.t ib m/ on 2 D C C Œ0; 1, for m ޚ. Since .t; b/ S for t Œ0; 1 according to (4.5), the inequality 2 2 1;b 2 in (4.12) gives 1 hm.t/ : j j Kk=2.m2 1/k=2 1 C LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 35

2 k=2 As we have seen, ޚ 1=.m 1/ < for k > 1, so ޚ hm.t/ con- m C 1 m verges uniformly on Œ02; 1. By (4.11) and (4.14), therefore, we see2 that P P 1 1 2nb 2nb an.k/ hm.t/e dt e hm.t/ dt D D Z0 m ޚ m ޚ Z0 X2 X2 1 e 2int e2nb dt: (4.16) D .t ib m/k m ޚ Z0 X2 C C By the change of variables x t m, we get D C 1 e 2int dt m 1 e 2in.x m/ m 1 e 2inx C dx C dx; k k k 0 .t ib m/ D m .x ib/ D m .x ib/ Z C C Z C Z C so 1 e 2int dt .t ib m/k m ޚ Z0 X2 C C m 1 e 2inx C dx D .x ib/k m ޚ Zm X2 C m 1 2inx m 1 2inx 1 e 1 e C dx C dx k k D m .x ib/ C m .x ib/ m 0 Z m 1 Z XD C XD C e 2inx 0 e 2inx e 2inx 1 dx 1 dx: (4.17) k k k D 0 .x ib/ C .x ib/ D .x ib/ Z C Z1 C Z1 C (Note that the integrals on the last line are ﬁnite for k > 1, since

e 2inx 1 (4.18) ˇ.x ib/k ˇ D .x2 b2/k=2 ˇ C ˇ C ˇ ˇ and the map x 1=.x2 ˇ b2/ lies inˇ L1.ޒ; dx/ for 2 > 1.) From (4.16) and ‘ Cˇ ˇ (4.17), we have 2inx 2nb 1 e an.k/ e dx: (4.19) D .x ib/k Z1 C In particular, dx a .k/ e2nb ; n 2 k=2 j j ޒ bk x 1 Z b C by (4.18). Setting t x=b, we can rewrite the right-hand side as D e2nb bdt e2nbc k ; (4.20) bk ޒ .t 2 1/k=2 D bk 1 Z C 36 FLOYDL.WILLIAMS

def where c dt=.t 2 1/k=2 < for k > 1. Since b > 0 is arbitrary, we let k D ޒ C 1 b . For n 0 or for n < 0, we see by the inequality (4.20) that an.k/ 0. ! 1 R D D Also for n 1; 2; 3; 4;::: and k > 1, it is known that D 2inx k e 2nb . 2i/ k 1 dx e n (4.21) ޒ .x ib/k D .k 1/! I Z C see the Remark below. The proof of Proposition 4.15 is therefore completed by way of equation (4.19). ˜

REMARK. Equation (4.21) follows from a contour integral evaluation: ib e 2iz .2/k k 1e ki=2 1C dz (4.22) k ib z D .k/ Z1C where ; b > 0, k > 1. The left-hand side here is e 2i.x ib/ e 2ix 1 C dx e2b 1 dx: .x ib/k D .x ib/k Z1 C Z1 C 2ix k 2b k k 1 Thus we can write 1 e dx=.x ib/ e . 2i/ =.k 1/! C D for ; b > 0 and k >11 an integer. The choice n (n 1; 2; 3; 4;::: ) gives R D D (4.21). For n ޚ, define 2 def 1 n.z/ k.nz/ D D .nz m/k m ޚ X2 C (see (4.11) for the last equality) on C. In definition (4.4), the n 0 contri- k k D k bution to the sum is m ޚ 0 1=m m1 1 1=m m1 1 1=. m/ k 2 f g D D C D D 2 m1 1 1=m (since k is even) 2.k/: Thus we can write Gk.z/ 2.k/ D P k D P P D C n ޚ 0 m ޚ 1=.m nz/ 2.k/ n1 1 n.z/ n1 1 n.z/. But P2 f g 2 C D C D C D n.z/ n.z/, again because k is even (the easy verification is left to the P D P P P reader). Therefore

1 1 G .z/ 2.k/ 2 n.z/ 2.k/ 2 .nz/ k D C D C k n 1 n 1 XD XD k 2.2i/ 1 1 2.k/ mk 1e2imnz D C .k 1/! n 1 m 1 XD XD (by Proposition 4.15, for k even), leading to

k 2.2i/ 1 2inz Gk.z/ 2.k/ k 1.n/e ; D C .k 1/! n 1 XD by formula (D.8) of Appendix D. This proves: LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 37

THEOREM 4.23. The holomorphic Eisenstein series G .z/, k 4; 6; 8; 10; 12; k D : : : , defined in (4.4) satisfy conditions (M1)0, (M1)00, and are holomorphic at infinity. In fact, G .z/ has Fourier expansion (4.2), where a0 2.k/ and k D k k 2.2i/ def 2.2i/ k 1 an k 1.n/ d D .k 1/! D .k 1/! d>0 dXn j for n 1. The G .z/ are therefore modular forms of weight k. k Since k is even in Theorem 4.23, formula (2.1) applies to .k/. As mentioned, a modular form is a cusp form if its initial Fourier coefficient a0 in equation (4.2) vanishes. By Theorem 4.23 the Gk.z/, for example, are not cusp forms since .k/ 0 for k even, k 4. In fact we know (by Theorem 3.17) ¤ that since .s/ is given by an Euler product, it is nonvanishing for Re s > 1. We return now to the discussion of Hecke L-functions, where we begin with results on estimates of Fourier coefficients of modular forms. The Gk .z/ already provide the example of how the general estimate looks. This involves only an estimate of the divisor function .n/ for > 1, where we first note that d > 0 runs through the divisors of n ޚ, n 1, as does n=d: 2 def n 1 1 .n/ d n n n ./: D D d D d d D 0

Therefore for an the n-th Fourier coefﬁcient of Gk.z/, Theorem 4.23 gives, for k k 1 k 1 n 1, an 2.2/ =.k 1/!.k 1/n C.k/n , where we have set j j D k def 2.2/ C.k/ .k 1/: D .k 1/! In general:

THEOREM 4.24. For a modular form f.z/ of weight k 4; 6; 8; 10;:::; with D Fourier expansion given by equation (4.2), there is a constant C.f; k/ > 0, k 1 depending only on f and k, such that an < C.f; k/n for n 1. If f.z/ is j j k=2 a cusp form, C.f; k/ > 0 can be chosen so that an < C.f; k/n for n 1. j j The idea of the proof is to ﬁrst establish the inequality

k=2 an < C.f; k/n ; n 1; (4.25) j j for a cusp form f.z/ by estimating f.z/ on a fundamental domain F C for the action of on C (given by restriction of the action of G in equation 38 FLOYDL.WILLIAMS

(4.3) to ), whence an estimate of f.z/ on C is readily obtained. Then an is estimated from the formula 1 2in.t ib/ an f.t ib/e dt (4.26) D C C Z0 for any b > 0; compare formula (4.14). By these arguments one can discover, in fact, that a modular form f.z/ of weight k is a cusp form if and only if there is a constant M.f; k/ > 0, depending only on f and k, such that

f.z/ < M.f; k/.Im z/ k=2 (4.27) j j on . Once (4.25) is established for a cusp form, the weaker result an < C j j C.f; k/nk 1, n 1, for an arbitrary modular form f.z/ of weight k follows from the fact that it holds for Gk .z/ (as shown above), and the fact that f.z/ differs from a cusp form (where one can apply the inequality (4.25)) by 2inz a constant multiple of Gk .z/. In fact, write f.z/ a0 n1 1 ane , 2inz D C D Gk.z/ b0 n1 1 bne (by equation (4.2)) where b0 2.k/, for exam- D C D def D P ple (by Theorem 4.23). Then f0.z/ f.z/ .a0=b0/ G .z/ is a modular form D k of weight k, withP Fourier expansion

1 2inz a0 1 a0 2inz f0.z/ a0 ane b0 bne D C b0 b0 n 1 n 1 XD XD 1 a0bn 2inz an e ; D b0 n 1 XD which shows that f0.z/ is a cusp form such that

a0 f.z/ f0.z/ Gk.z/: (4.28) D C b0 To complete our sketch of the proof of Theorem 4.24, details of which can be found in section 6.15 of [2], for example, we should add further remarks regarding F. By deﬁnition, a fundamental domain for the action of on C is an open set F such that (F1) no two distinct points of F lie in the same C -orbit: if z1; z2 F with z1 z2, then there is no such that z1 z2. 2 ¤ 2 D We also require condition: (F2) given z , there exists some such 2 C 2 that z F (= the closure of F). The standard fundamental domain, as is 2 N well-known, is given by

def F z z > 1; Re z < 1 ; (4.29) D 2 C j j j j j 2 and is shown at the the top˚ of the next page. « LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 39

F is this interior region

x 1 0 1 2 2

If Mk . / and Sk. / denote the space of modular forms and cusp forms of weight k 4; 6; 8; 10;::: , respectively, there is the ރ-vector space direct sum D decomposition M . / S . / ރG ; (4.30) k D k ˚ k by (4.28). The sum in (4.30) is indeed direct since (as we have seen) Gk.z/ is not a cusp form. If f M . / with Fourier expansion (4.2), the corresponding Hecke L- 2 k function L.s f/ is given by deﬁnition (3.21): I

def 1 a L.s f/ .s/ n : (4.31) I D f D ns n 1 XD By Theorem 4.24 this series converges absolutely for Re s > k, and for Re s > 1 k if f S . /. On these respective domains L.s f/ is holomorphic in s (by C 2 2 k I an argument similar to that for the Riemann zeta function), as we have asserted in Lecture 3. Since Theorem 3.25 is based on equation (3.24), which is based on equation (3.23), our proof of it actually shows the following reformulation:

THEOREM 4.32. Suppose the Fourier coefﬁcients an of f M . / satisfy the 2 k multiplicative condition (3.23), with at least one an nonzero:

k 1 an1n2 an1 an2 d (4.33) D d 2 d>0 d nX1; d n2 j j 40 FLOYDL.WILLIAMS for n1; n2 1. Then L.s f/ has the Euler product representation I 1 L.s f/ (4.34) I D 1 pk 1 2s a p s p P p Y2 C for Re s > k. If f S . /, equation (4.34) holds for Re s > 1 k=2. 2 k C Here, we only need to note that a1 1. For if some an 0, then by (4.33) 2 D ¤ ana1 an 1=1 an and so a1 1. D D D Theorem 4.32 raises the question of finding modular forms whose Fourier coefficients satisfy the multiplicative condition (3.23) = condition (4.33). This question was answered by Hecke (in 1937), who found, in fact, all such forms. As was observed in Lecture 3, the multiplicative condition is satisfied by normalized simultaneous eigenforms: nonzero forms f.z/ with a1 1, that are D simultaneous eigenfunctions of all the Hecke operators T .n/; n 1; see definition (3.22). More concretely, among the non-cusp forms the normalized simultaneous eigenforms turn out to be the forms .k 1/! f.z/ Gk.z/; D 2.2i/k where indeed, by Theorem 4.23, a1 k 1.1/ 1. D D We mention that the Hecke operators T .n/ n 1 map the space Mk . / to f g itself, and also map the space Sk. / to itself. For f Mk . / with Fourier 2inz 2 expansion f.z/ 1 ane on , as in (4.2), .T .n/f /.z/ has Fourier D n 0 C expansion D P 1 .T .n/f /.z/ a.n/e2imz D m m 0 XD .n/ .n/ k 1 on C, where a0 a0k 1.n/ and am 0 k. Here the integral

def 1 s 1 J .s/ f.it/ a0 t dt (4.36) f D Z1 LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 41 is an entire function of s. Notice that equation (4.35) is similar in form to (1.12). Again since 1= .s/ is an entire function, and since .1=s/.1= .s// D 1= .s 1/, we can write equation (4.35) as C .2/s .2/s a i k .2/sa L.s f/ J .s/ i kJ .k s/ 0 0 (4.37) I D .s/ f C f C .s/ .s k/ .s 1/ C for Re s > k. If f.z/ is a cusp form, we see that L.s f/ extends to an entire I function by way of the first term in (4.37). In general, we see that L.s f/ I extends meromorphically to ރ, with a single (simple) pole at s k with residue k k k k D .2/ a0i = .k/ .2/ a0i =.k 1/!. Also by equation (4.37), for k s k D ¤ (i.e., s 0), we have ¤ i k .2/s k.k s/L.k s f/ I k s k .2/ a0i a0.k s/ i k .2/s k.k s/ J .k s/ i kJ .s/ D .k s/ f C f C s .k s 1/ C a i k a i k J .k s/ J .s/ 0 0 D f C f C s k s (again since .w 1/ w .w/, and since i 2k 1 for k even); the right-hand C D D side in turn equals .2/ s.s/L.s f/. That is, I .2/ s.s/L.s f/ i k .2/s k .k s/L.k s f/ (4.38) I D I for s 0, which is the functional equation for L.s f/, which compares with ¤ I the functional equation for the Riemann zeta function; see [17; 18]. The Eisenstein series Gk.z/ can be used as building blocks to construct other modular forms. It is known that any modular form f.z/ is, in fact, a finite sum n m of the form f.z/ n;m 0 cnmG4.z/ G6.z/ for suitable complex numbers D cnm. Of particular interest are the discriminant form P def 3 2 .z/ .60G4.z// 27 .140G6.z// (4.39) D and the modular invariant def 3 J.z/ .60G4.z// =.z/ (4.40) D which is well-defined since it is true that .z/ never vanishes on C. .z/ is a modular form of weight 12, since if f1.z/;f2.z/ are modular forms of weight k1; k2, then f1.z/f2.z/ is a modular form of weight k1 k2. Similarly J.z/ is C a weak modular form of weight k 0: J. ; z/ J.z/ for , z . The D D 2 2 C form J.z/ was initially constructed by R. Dedekind in 1877, and by F. Klein in 1878. Associated with it is the equally important modular j-invariant def j.z/ 1728J.z/: (4.41) D 42 FLOYDL.WILLIAMS

.z/ is connected with the Dedekind eta function .z/ (see deﬁnition (3.27)) by the Jacobi identity .z/ .2/12.z/24; (4.42) D which with equation (3.28) shows that .z/ has Fourier expansion

.z/ .2/12 1 .n/e2inz; (4.43) D n 1 PD where .n/ is the Ramanujan tau function, and which in particular shows that .z/ is a cusp form: .z/ S12. /, which can also be proved directly by 2 definition (4.39) and Theorem 4.23, for k 4; 6. There are no nonzero cusp D forms of weight < 12. Note that by Theorem 4.24, there is a constant C > 0 such that .n/ < C n6 j j for n 1. However P. Deligne proved the Ramanujan conjecture .n/ 11=2 j j 0.n/n for n 1, where 0.n/ is the number of positive divisors of n. As we remarked in Lecture 3, the .n/ (remarkably) are all integers. This can be proved using definition (4.39) and Theorem 4.23, for k 4; 6. It is also true D that, thanks to the factor 1728 in definition (4.41), all of the Fourier coefficients of the modular j-invariant are integers:

2iz 1 2inz j.z/ 1e ane (4.44) D C n 0 D with each an ޚ: P 2 a0 744; a1 196;884; a2 21;493;760; a3 864;299;970; D D D D a4 20;245;856;256; a5 333;202;640;600; ::: (4.45) D D An application of the modular invariant j.z/, and of the values in (4.45), to three-dimensional gravity with a negative cosmological constant will be given in my Speaker’s Lecture; see especially equation (5-8) on page 343 and the subsequent discussion. In the deﬁnition (4.4) of the holomorphic Eisenstein series Gk .z/, one cannot take k 2 for convergence reasons. However, Theorem 4.23 provides a D suggestion of how one might proceed to construct a series G2.z/. Namely, take k 2 there and thus deﬁne D 2 def 2 1 2inz 2 1 2inz G2.z/ 2.2/ 2.2i/ .n/e 8 .n/e (4.46) D C n 1 D 3 n 1 D D def P def P on C, where .n/ 1.n/ 0

2inz 1 2nB Im z B > 0 on K, so .n/e 2 n.n 1/e on K, and again 2nB j j C 2inz n1 1 n.n 1/e < for B > 0. By the M -test, n1 1 .n/e con- D C 1 D verges uniformly on K, which (by the Weierstrass theorem) means that G2.z/ P P is a holomorphic function on C. Another expression for G2.z/ is 1 G2.z/ 2.2/ : (4.47) D C .m nz/2 n ޚ 0 m ޚ 2Xf g X2 C To check this, start by taking k 2 in deﬁnition (4.11) and in Proposition 4.15: D 1 1 2 1 2ikz 2.z/ . 2i/ ke : (4.48) z2 C .z m/2 D D m ޚ 0 k 1 2Xf g C XD Replace z by nz in (4.48) and sum on n from 1 to : 1 1 1 1 1 1 .2/ . 2i/2 ke2iknz: (4.49) z2 C .nz m/2 D n 1 m ޚ 0 n 1 k 1 XD 2Xf g C XD XD By (D.6) (see page 90), we obtain 1 .n/ d.k; n/k: (4.50) D k 1 XD def 2inz For an e , n 1, z , and for k 1 ﬁxed the series 1 d.k; n/an D 2 C n 1 clearly converges absolutely, since Im z > 0 and 0 d.k; n/ D1. Then the P series n1 1 akn converges and equals n1 1 d.k; n/an, by the Scholium of AppendixD (page 91): D P P 1 1 e2iknz d.k; n/e2inz (4.51) D n 1 n 1 XD XD which gives, by equation (4.50)

1 1 1 1 1 .n/e2inz d.k; n/e2inz k d.k; n/e2inz D D n 1 n 1 k 1 k 1 n 1 XD XD XD XD XD 1 1 1 1 k e2iknz ke2iknz; (4.52) D D k 1 n 1 n 1 k 1 XD XD XD XD which in turn allows for the expression

2 1 1 1 .2/ 2 2.2i/2 .n/e2inz z2 C .nz m/2 D n 1 m ޚ 0 n 1 XD 2Xf g C XD G2.z/ 2.2/ (4.53) D 44 FLOYDL.WILLIAMS by equation (4.49), provided the commutations of the summations over k; n in (4.52) are legal. But for y Im z, D 1 1 2inz 1 1 2ny 1 2ny d.k; n/ke d.k; n/ke .n/e n 1 k 1 j jD n 1 k 1 D n 1 D D D D D P P P P 2inz P by (4.50), and this equals 1 .n/e , which is finite, as we have seen. n 1 j j This justifies the first commutation.D Similarly, P e 2ky k 1 1 ke2iknz 1 k 1 .e 2ky/n 1 k 1 2ky 2ky k 1 n 1 j jDk 1 n 1 Dk 1 1 e Dk 1 e 1 D D D D D D isP finiteP by the integralP test:P P P t dt 1 u du 1 1 < ; (4.54) 2yt 2 u 1 e 1 D .2y/ 2y e 1 1 Z Z as we shall see later by Theorem 6.1, for example. This justifies the second commutation in equation (4.52). Since for n 1 1 1 1 ; (4.55) .m nz/2 D . m nz/2 D .m nz/2 m ޚ 0 m ޚ 0 m ޚ 0 2Xf g 2Xf g 2Xf g C the double sum on the right-hand side of (4.47) can be written as

1 1 1 .nz/2 C .m nz/2 n ޚ 0 m ޚ 0 2Xf g 2Xf g C 1 1 1 1 1 1 2 D n2z2 C .m nz/2 C .m nz/2 n 1 n 1 m ޚ 0 n 1 m ޚ 0 XD XD 2Xf g C XD 2Xf g .2/ 1 1 2 2 ; (4.56) D z2 C .m nz/2 n 1 m ޚ 0 XD 2Xf g C which is the left-hand side of (4.53). That is, G2.z/ 2.2/ equals the double sum on the right-hand side of (4.47), proving (4.47). Using equation (4.47) one can eventually show that G2.z/ satisﬁes the rule

1 2 G2 z G2.z/ 2iz: (4.57) z D Because of the term 2iz in (4.57), G2.z/ is not a modular form of weight 2. That is, condition (M1)00 above is not satisﬁed, although condition (M1)0 is: G2.z 1/ G2.z/ by deﬁnition (4.46). Equation (4.57) also follows by a C D transformation property of the Dedekind eta function, whose logarithmic derivative turns out to be a constant multiple of G2.z/. Thus we indicate now an alternative derivation of the rule (4.57). LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 45

def On the domain D w ރ w < 1 , the holomorphic function 1 w D f 2 j j j g C is nonvanishing and it therefore has a holomorphic logarithm g.w/ that can def be chosen so as to vanish at w 0: eg.w/ 1 w on D. In fact g.w/ n D def 2iz D C D n1 1. w/ =n. Again for q.z/ e , z C, consider the n-th partial D def n k D 2 sum sn.z/ g. q.z/ /, which is well-deﬁned, because q.z/ D for P D k 1 2 z implies qD.z/k D for k > 0. We claim that the series 2 C P 2

def 1 k .z/ g. q.z/ / lim sn.z/ (4.58) D D n k 1 !1 XD k n converges. We have .z/ 1 1 .q.z/ / =n, where for y Im z D k 1 n 1 D (again) D D P P k n 1 1 .q.z/ / 1 1 1 .e 2ny/k n D n n 1 k 1 ˇ ˇ n 1 k 1 XD XD ˇ ˇ XD XD ˇ ˇ 2ny ˇ ˇ 1 1 e 1 1 1 ˇ ˇ ; D n 1 e 2ny D n e2ny 1 n 1 n 1 XD XD which is finite by the integral test; compare with (4.54), for example. This allows k n k n us to write n1 1 k1 1.q.z/ / =n k1 1 n1 1.q.z/ / =n, and shows D D D D D n the finiteness of these series. Hence .z/ is finite. Now n1 1.1 q.z/ / nP P k Pn gP. q.z/k / D D limn k 1.1 q.z/ / limn k 1 e , by the definition of !1 D !1 g.w/, and thisD equals e .z/ by (4.58). ThatD is, for the DedekindQ eta function Q Q

1 .z/ eiz=12 .1 q.z/n/ (4.59) D n 1 YD on C deﬁned in (3.27) we see that

.z/ eiz=12 .z/: (4.60) D def Differentiation of the equation eg.w/ 1 w gives g .w/ 1=eg.w/ 1=.1 w/ D C 0 D D C (of course), which with termwise differentiation of (4.58) (whose justiﬁcation we skip) gives

k 1 k k 1 1 kq.z/ 0.z/ g0. q.z/ /. kq.z/ q0.z// 2i D D 1 q.z/k k 1 k 1 XD XD 1 1 1 2i k .q.z/k/n 2i .n/e2inz; D D k 1 n 1 n 1 XD XD XD 46 FLOYDL.WILLIAMS by equation (4.52). Therefore, by (4.60), we obtain

i 1 2inz 0.z/ .z/ 2i .n/e ; D 12 n 1 D or P .z/ G .z/ 0 2 (4.61) .z/ D 4i by definition (4.46). Now .z/ satisfies the known transformation rule 1 e i=4pz.z/; z D where we take arg z . ;/. Differentiation gives 2 1 1 i=4 pz e pz0.z/ .z/ 0 2 C 2z .z/ 1 z z 0 (4.62) 1 1 D D .z/ C 2z z z 1 2 1 which by equation (4.61) says that G2 =4iz G2.z/=4i . This z D C 2z is immediately seen to imply the transformation rule (4.57). In the lectures of Geoff Mason and Michael Tuite the particular normaliza- k tion Gk.z/=.2i/ of the Eisenstein series is considered, which they denote by Ek.z/. In particular, by definition (4.46),

1 1 2inz E2.z/ 2 .n/e : D 12 C n 1 XD However, other normalized Eisenstein series appear in the literature that also might be denoted by Ek.z/. For example, in [20] there is the normalization def (and notation) E .z/ G .z/=2.k/. k D k Lecture 5. Dirichlet L-functions

Equation (3.19) has an application to Dirichlet L-functions, which we now consider. To construct such a function, we need first a character modulo m, where m > 0 is a fixed integer. This is defined as follows. Let Um denote the def group of units in the commutative ring ޚ ޚ=mޚ. Thus if n n mޚ .m/ D N D C denotes the coset of n ޚ in ޚ , we have n Um a ޚ such 2 .m/ N 2 ()9N 2 .m/ that an 1. One knows of course that n Um .n; m/ 1 (i.e. n and N N D N N 2 () D m are relatively prime). A character modulo m is then (by definition) a group def homomorphism Um ރ ރ 0 . For our purpose, however, there is an W ! D f g LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 47 equivalent way of thinking about characters modulo m. In fact, given , define ޚ ޚ ރ by W ! def .n/ if .n; m/ 1, ޚ.n/ N D (5.1) D 0 otherwise, for n ޚ. For n; n1; n2 ޚ, ޚ satisfies: 2 2 (D1) ޚ.n/ 0 .n; m/ 1; D () ¤ ޚ (D2) ޚ.n1/ ޚ .n2/ when n1 n2 in ; D N D N .m/ (D3) ޚ.n1n2/ ޚ.n1/ ޚ.n2/ when .n1; m/ 1 and .n2; m/ 1. D D D Conversely, suppose 0 ޚ ރ is a function that satisfies the three conditions W ! (D1), (D2), and (D3). Define Um ރ by .n/ 0.n/ for n ޚ such that W ! N D 2 .n; m/ 1. The character is well-defined by (D2), and Um ރ by D W ! (D1). By (D3), .ab/ .a/ .b/ for a; b Um, so we see that is a character D 2 modulo m. Moreover the induced map . 0/ޚ ޚ ރ given by definition (5.1) W ! coincides with 0. Note that ޚ is completely multiplicative:

(D4) ޚ.n1n2/ ޚ.n1/ ޚ.n2/ for all n1; n2 ޚ. D 2 For if either .n1; m/ 1 or .n2; m/ 1, then .n1n2; m/ 1, so that by (D1) ¤ ¤ ¤ both ޚ.n1/ ޚ.n2/ and ޚ.n1n2/ are zero. If both .n1; m/ 1 and .n2; m/ 1, D D then already ޚ.n1n2/ ޚ.n1/ ޚ.n2/ by (D3). D Note also that since .a/ ރ for every a Um (that is, .a/ 0), we have 2 2 ¤ 0 .1/ .11/ .1/ .1/ by (D4), so .1/ 1. Moreover since .1; m/ 1, ¤ N D N N D N N N D D ޚ.1/ .1/, by (5.1), which in turn equals 1. D N One ﬁnal property of ޚ that we need is:

(D5) .a/ 1 for all a Um; hence ޚ.n/ 1 for all n ޚ. j jD 2 j j 2 The proof of (D5) makes use of a little theorem in group theory which says G that if G is a ﬁnite group with G elements, then aj j 1 for every a G. Now, j j D Um 2Um given a Um, we can write (as just seen) 1 .1/ .a / .a/ (since 2 D N D j j D j j is a group homomorphism), which shows that .a/ is a Um -th root of unity: j j .a/ 1 for all a Um. Hence ޚ.n/ 1 for all n ޚ, by deﬁnition (5.1). j jD 2 j j 2 Given a character modulo m, it follows that we can form the zeta function, or Dirichlet series

def 1 ޚ.n/ .n/ L.s; / N ; (5.2) D ns D ns n 1 .n;m/ 1 XD XD called a Dirichlet L-function, which converges for Re s > 1, by (D5). L.s; / is holomorphic on the domain Re s > 1, by the same argument given for the Rie- mann zeta function .s/. Since ޚ 0 ( ޚ.1/ 1), and since ޚ is completely 6 D multiplicative, formula (3.19) implies: 48 FLOYDL.WILLIAMS

THEOREM 5.3 (EULER PRODUCT FOR DIRICHLET L-FUNCTIONS). Assume Re s > 1. Then 1 1 L.s; / : (5.4) s s D 1 ޚ.p/p D 1 ޚ.p/p p P p P Q2 pQ2- m The second statement of equality follows by (D1), since for a prime p P, 2 saying that .p; m/ 1 is the same as saying that p m. ¤ j As an example, define 0 ޚ ރ by W ! 1 if .n; m/ 1, 0.n/ D D 0 otherwise, for n ޚ. Then 0 satisfies (D1). If n1; n2; l ޚ such that n1 n2 lm (i.e., 2 2 D C n1 n2/, then .n1; m/ 1 .n2; m/ 1, so 0 satisfies (D2). If .n1; m/ 1 N D N D () D D and .n2; m/ 1, then .n1n2; m/ 1, so 0 also satisfies (D3), and 0 therefore D D defines a Dirichlet character modulo m. We call 0 (or the induced character Um ރ ) the principal character modulo m. Again since p is a prime, we see ! by equation (5.4) that for Re s > 1 1 L.s; 0/ s : (5.5) D .1 p / p P pQ2- m Then for Re s > 1 1 1 L.s; 0/ s s .s/ (5.6) .1 p / D .1 p / D p P p P p2m 2 Qj Q by formula (0.2). That is,

s L.s; 0/ .s/ .1 p / (5.7) D p P pY2m j for Re s > 1. If 0 Um ރ also denotes the character modulo m induced by 0 ޚ ރ, W ! W ! then 0.n/ 1 for every n Um (by (5.1)) since .n; m/ 1. N D N 2 D As another simple example, take m 5: Z.5/ 0; 1; 2; 3; 4 , and it is D D fN N N N defNg def easily checked that U5 1; 2; 3; 4 . Moreover the equations .1/ .4/ 1, def def DfN N N Ng .5/ N D N D .2/ .3/ 1 define a character U5 ރ modulo 5. The N D N D D W ! induced map ޚ ޚ ރ in definition (5.1) is given by ޚ.1/ 1, ޚ.2/ 1, W ! D D ޚ.3/ 1, ޚ.4/ 1, ޚ.5/ 0 (since .5; 5/ 1), ޚ.6/ 1, ޚ.7/ 1, D D D ¤ D D ޚ.8/ 1, ޚ.9/ 1, . . . (since 6 1, 7 2, 8 3, 9 4, with .n; 5/ 1 D D N D N N D N N D N N D N D LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 49 for n 6; 7; 8; 9). The corresponding Dirichlet L-function is therefore given, D for Re s > 1, by 1 1 1 1 1 1 1 L.s; .5// 1 ; D 2s 3s C 4s C 6s 7s 8s C 9s ˙ Next take m 8 Z 0; 1; 2; 3; 4; 5; 6; 7 , U 1; 3; 5; 7 : The map .8/ N N N N N N N N 8 N N N N .8/ D W Df def def g def Df def g U8 ރ given by .1/ .7/ 1, .3/ .5/ 1 is a character D W ! N D N D N D N D modulo 8, with ޚ ޚ ރ in definition (5.1) given by ޚ.1/ 1, ޚ.2/ 0, W ! D D ޚ.3/ 1, ޚ.4/ 0, ޚ.5/ 1, ޚ.6/ 0, ޚ.7/ 1, ޚ.8/ 0, D D D D D D ޚ.9/ 1, ޚ.10/ 0, ޚ.11/ 1, ޚ.12/ 0, ޚ.13/ 1, ޚ.14/ 0, D D D D D D ޚ.15/ 1, ޚ.16/ 0, ޚ.17/ 1, . . . . Then D D D 1 1 1 1 1 1 1 1 L.s; .8// 1 : D 3s 5s C 7s C 9s 11s 13s C 15s C 17s ˙ From formula (5.7) it follows that the L-function L.s; 0/ admits a meromorphic continuation to the full complex plane, with s 1 as its only singularity — 1 D a simple pole with residue 1 . If 0 it is known that L.s; / p P; p m p ¤ at least extends to Re s > 0 and,2 moreover,j that L.1; / 0. For example, for the Q ¤ characters .5/, .8/ modulo 5 and 8, respectively, constructed in the previous examples, one has 1 3 p5 1 L.1; .5// log C ; L.1; .8// log.3 2p2/: D p 2 D p C 5 8 If 0 is a primitive character modulo m, a notion that we shall define ¤ presently, then L.s; / does continue meromorphically to ރ, and it has a decent functional equation. First we define the notion of an imprimitive character. Suppose k > 0 is a divisor of m. Then there is a natural (well-defined) map q ޚ=mޚ ޚ=kޚ, given by def W ! q.n mޚ/ n kޚ for n ޚ. If .n; m/ 1 then .n; k/ 1 since k m. C D C def2 D D j Therefore the restriction q q Um maps Um to Uk, and is a homomorphism D j .k/ between these two groups. Let Uk ރ be a character modulo k. By .k/ W ! .k/ definition, is a homomorphism and hence so is q Um ރ. That ı W def! is, given a positive divisor k of m we have an induced character .k/ q D ı modulo m. Characters modulo m that are induced this way, say for k m, ¤ are called imprimitive. is called a primitive character if it is not imprimitive, in which case m is also called the conductor of . Thus for a primitive character modulo m, the L-function L.s; / satisfies a theory similar to (but a bit more complicated than) that of the Riemann zeta function .s/. In the Introduction we referred to the prime number theorem, expressed in equation (0.3), as a monumental result, and we noted quite briefly the role of .s/ in its proof. Similarly, the study of the L-functions L.s; / leads to a 50 FLOYDL.WILLIAMS monumental result regarding primes in an arithmetic progression. Namely, in 1837 Dirichlet proved that there are infinitely many primes in any arithmetic progression n, n m, n 2m, n 3m, : : : , where n; m are positive, relatively C C C prime integers — a key aspect of the proof being the fact (pointed out earlier) that L.1; / 0 if 0. Dirichlet’s proof relates, moreover, L.1; / to a Gauss- ¤ ¤ ian class number — an invariant in the study of binary quadratic forms. One can obtain, also, a prime number theorem for arithmetic progressions (from the Siegel–Walfisz theorem), where the counting function .x/ in (0.3) is replaced def by the function .x m; n/ the number of primes p x, with p n.mod m/, I D for n; m relatively prime. One can also formulate and prove a prime number theorem for graphs. This is discussed in section 3.3 of the lectures of Audrey Terras.

Lecture 6. Radiation density integral, free energy, and a ﬁnite-temperature zeta function

Theorems 1.13 and 1.18 provide for integral representations of .s/, for Re s > 1, that serve as starting points for its analytic continuation. The following, nice integral representation also serves as a starting point. We apply it to compute Planck’s radiation density integral. We also consider a free energy – zeta function connection.

THEOREM 6.1. For Re s > 1 1 t s 1 dt .s/ 1 : (6.2) D .s/ et 1 Z0 We can regard the integral on the right as the sum

1 t s 1 dt t s 1 dt 1 ; et 1 C et 1 Z0 Z1 where the second integral converges absolutely for all s ރ, and the ﬁrst inte- 1 s 1 t 2 gral, understood as the limit lim˛ 0C ˛ t dt=.e 1/, exists for Re s > 1. ! The proof of Theorem 6.1 is developedR in two stages. First, for ˛;ˇ; a ޒ and ˇ aˇ 2 s ރ, with ˛ < ˇ and a > 0, write e at t s 1 dt a s e vvs 1dv, by 2 ˛ D a˛ the change of variables v at. In particular, D R R ˇ aˇ at s 1 s t s 1 e t dt a e t dt for ˇ > 1; (6.3) 1 D a Z 1 Z a e at t s 1 dt a s e t t s 1 dt for ˛ < 1. (6.4) D Z˛ Za˛ LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 51

For s ރ; m > 0, consider the integral 2 s 1 mt t s 1 mt def 1 t e e 1 t e Im.s/ dt dt (6.5) D 1 e t D et 1 Z0 Z0 which we check does converge for Re s > 1. For t > 0, we have et > 1 t, so C 1=.et 1/ < 1=t, and hence t s 1e mt t 1e mt < t 2e mt (6.6) et 1 t D ˇ ˇ ˇ ˇ ˇ ˇ 2ˇ mt . 1/ mˇ t 2 for Re s. By (6.3),ˇ 1 t ˇe dt m m e t dt for ˇ > 1. D 2 mt D Let ˇ : then 1 t e dt exists and ! 1 1 R R R 1 1 t 2e mt dt 1 e t t 2 dt: D m 1 Z1 Zm t s 1e mt In view of (6.6), therefore, dt converges absolutely for every 11 et 1 s ރ and m > 0, and 2 R t s 1e mt t s 1e mt 1 1 dt 1 dt 1 e t t 2 dt: (6.7) et 1 et 1 m 1 ˇ Z1 ˇ Z1 ˇ ˇ Zm ˇ ˇ ˇ ˇ Byˇ the change of variablesˇ v ˇ 1=t for t >ˇ 0, ˇ ˇ Dˇ ˇ s 1 1 m.1=v/ 1 t s 1e mt 1=˛ e dt v dv (6.8) t 1=v 2 ˛ e 1 D 1 .e 1/v Z Z for 0 < ˛ < 1. Here, by the inequality in (6.6), we can write

s 1 e m.1=v/ 2 e m=v 1 1 m=v < v e v : (6.9) ˇ v .e1=v 1/v2 ˇ v v2 D ˇ ˇ ˇ ˇ ˇ 1=˛ 1ˇ 1=˛ ˇ .1=˛/ ˇ 1 1 But v dv , so lim v dv for > 1, i.e., 1 D 1 ˛ 0C 1 D 1 Z ! Z s 1 1 m.1=v/ e 1 v dv 1=v 2 1 .e 1/v Z converges absolutely for Re s > 1 and

s 1 s 1 1 m.1=v/ 1 m.1=v/ e e 1 v dv 1 v dv 1 v e m=vdv; 1=v 2 1=v 2 ˇ 1 .e 1/v ˇ 1 ˇ .e 1/v ˇ 1 ˇ Z ˇ Z ˇ ˇ Z ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ 52 FLOYDL.WILLIAMS

ˇ m=v by the inequality in (6.9). The right-hand side equals limˇ 1 v e dv, or, by the change of variables t 1=v, !1 D R 1 m 2 mt . 1/ t 2 lim t e dt lim m e t dt ˇ 1=ˇ D ˇ m=ˇ !1 Z !1 Z 1 m e t t 2 dt: D m 1 Z0 1 s 1 mt t e That is, by (6.8), lim t dt exists for Re s > 1, and, with Re s, ˛ 0C ˛ e 1 D ! Z 1 t s 1e mt 1 m lim dt e t t 2 dt: (6.10) t 1 ˛ 0C ˛ e 1 m 0 ˇ ! Z ˇ Z ˇ ˇ We have thereforeˇ checked that the integralˇ Im.s/ deﬁned by (6.5) converges ˇ ˇ s 1 mt t for Re s > 1 (and in fact the portion 1 t e =.e 1/ dt converges abso- 1 lutely for all s ރ), and that, moreover, for Re s, we have 2 R D 1 t s 1e mt 1 m dt e t t 2 dt; et 1 m 1 ˇ Z0 ˇ Z0 ˇ ˇ ˇ t s 1e mt ˇ 1 ˇ 1 dtˇ 1 e t t 2 dt; et 1 m 1 ˇ Z1 ˇ Zm ˇ ˇ by the inequalitiesˇ (6.7) and (6.10).ˇ This says that ˇ ˇ m 1 t 2 1 t 2 Im.s/ e t dt e t dt m 1 C ˇ ˇ Z0 Zm ˇ ˇ ˇ ˇ 1 1 ˇ ˇ 1 e t t 2 dt . 1/; D m 1 D m 1 Z0 by deﬁnition (1.6). Since m 1 0 as m for > 1, we see that ! ! 1 lim Im.s/ 0 (6.11) m D !1 for Re s > 1 ! We move now to the second stage of the proof of Theorem 6.1, which is quite brief. Fix integers n; m with 1 n m and s ރ with Re s > 1, and set v nt. nt s 1 s 2v s 1 s D Again by (1.6), 1 e t dt n 1 e v dv .s/=n , so 0 D 0 D m m 1 R R e t .1 e mt / .s/ 1 t s 1 e nt dt 1 t s 1 dt ns D D 1 e t n 1 Z0 n 1 Z0 XD XD s 1 t 1 t e dt Im.s/: D 1 e t C Z0 LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 53

(Here the the last equality comes from (6.5) and the last but one from the formula for the partial sum of the geometric series.) Therefore

m s 1 1 1 t .s/ s t dt Im.s/: (6.12) n D 0 e 1 C n 1 Z XD We check the existence of the integral on the right-hand side for Re s > 1: write t s 1 t s 1e t t s 1e t et 1 D C et 1 s 1 t for t > 0; the integral 1 t e dt .s/ converges for Re s > 0 and the 0 D integral of the last term, which equals I .s/ by (6.5), converges for Re s > 1, as R 1 established in the first stage of the proof. s 1 1 t dt Let m in equation (6.12): then .s/.s/ t by (6.11) for ! 1 D 0 e 1 Re s > 1, concluding the proof of Theorem 6.1. Z COROLLARY 6.13. For a > 0, Re s > 1, t s 1 dt .s/.s/ 1 : (6.14) eat 1 D as Z0 In particular, for a > 0, n 1; 2; 3; 4;::: D 2n 1 n 1 2n t . 1/ .2=a/ B2n 1 dt C : (6.15) eat 1 D 4n Z0 PROOF. This follows from (6.2) once the obvious change of variables v at is D executed. By formula (6.14), t 2n 1 .2n/.2n/ 1 dt eat 1 D a2n Z0 for n ޚ, n 1. Since .2n/ .2n 1/! (because .m/ .m 1/! for 2 D D m ޚ; m 1), one can now appeal to formula (2.1) to conclude the proof of 2 equation (6.15). ˜ As an application of formula (6.15) we shall compute Planck’s radiation density integral. But first we provide some background. On 14 December 1900, a paper written by Max Karl Ernst Ludwig Planck and entitled “On the theory of the energy distribution law of the normal spectrum” was presented to the German Physical Society. That date is considered to be the birthday of quantum mechanics, as that paper set forth for the first time the hypothesis that the energy of emitted radiation is quantized. Namely, the energy cannot assume arbitrary values but only integral multiples 0; h; 2h; 3h;::: of the basic energy value E h, where is the frequency of the radiation and h D is what is now called Planck’s constant. We borrow a quotation from Hermann 54 FLOYDL.WILLIAMS

Weyl’s notable book [37]: “The magic formula E h from which the whole D of quantum theory is developed, establishes a universal relation between the frequency of an oscillatory process and the energy E associated with such a process”. The quantization of energy has profound consequences regarding the structure of matter. Planck was led to his startling hypothesis while searching for a theoretical justiﬁcation for his newly proposed formula for the energy density of thermal (or “blackbody”) radiation. Lord Rayleigh had proposed earlier that year a theoretical explanation for the experimental observation that the rate of energy emission f. T / by a body at temperature T in the form of electromagnetic radiation I of frequency grows, under certain conditions, with the square of , and the total energy emitted grows with the fourth power of T . In the quantitative form derived by James Jeans a few years later, Rayleigh’s formula reads

82 f. T / kT; (6.16) I D c3 where c is the speed of light and k is Boltzmann’s constant. As grows, however, this formula was known to fail. Wilhelm Wien had already proposed, in 1896, the empirically more accurate formula

f. T / a3e b=T : (6.17) I D Unlike the Rayleigh–Jeans formula (6.16), Wien’s avoids the “ultraviolet catastrophe”. (This colorful name was coined later by Paul Ehrenfest for the notion that a functional form for f. T / might yield an inﬁnite value for the I total energy, 1 f. T /d — “ultraviolet” because the divergence sets in 0 I D 1 at high frequencies.) However, the lack of a theoretical explanation for Wien’s R law, and its wrong prediction for the asymptotic limit at low frequencies — proportional to 3 rather than 2 — made it unsatisfactory as well. By October 1900 Planck had come up with a formula that had the right asymptotic behavior in both directions and was soon found to be very accurate:

82 h f. T / ; (6.18) I D c3 eh=kT 1 where the new constant h was introduced. In his December paper, already mentioned, he provides a justification for this formula using the earlier notions of electromagnetic oscillators and statistical-mechanical entropy, but invoking the additional assumption that the energy of the oscillators is restricted to multiples of E h. This then is the genesis of quantization. D Note that given the approximation ex 1 x for a very small value of x, one ' C has the low frequency approximation eh=kT 1 h=kT , which when used ' LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 55 in formula (6.18) gives f. T / 82kT =c3 — the Rayleigh–Jeans result I ' (6.16), as expected by our previous remarks. We now check that in contrast to an infinite total energy value implied by formula (6.16), integration over the full frequency spectrum via Planck’s law (6.18) does yield a finite value. The result is:

PROPOSITION 6.19. Planck’s radiation density integral 3 def 8h d I.T / 1f. T /d 1 3 h=kT D 0 I D c 0 e 1 Z Z (see (6.18)) has the ﬁnite value 85k4T 4=15c3h3. The proof is quite immediate. In formula (6.15) choose n 2, a h=kT ; then D D 8h 2kT 4 B I.T / . 1/3 4 : D c3 h 8 Since B4 1=30 by (2.2), the desired value of I.T / is achieved. D The radiation energy density f. T / in (6.18) is related to the thermody- I namics of the quantized harmonic oscillator; namely, it is related to the thermodynamic internal energy U.T /. We mention this because U.T /, in turn, is related to the Helmholtz free energy F.T / which has, in fact, a zeta function connection. A quick sketch of this mix of ideas is as follows, where proofs and details can be found in my book on quantum mechanics [42] (along with some historic remarks). One of the most basic, elementary facts of quantum mechanics is that the quantized harmonic oscillator of frequency has the sequence

def 1 En n 2 h n1 0 D C D as its energy levels. The corresponding˚ partition « function Z.T / is given by

def 1 E Z.T / exp n ; (6.20) D kT n 0 XD where again T denotes temperature and k denotes Boltzmann’s constant. This sum is easily computed:

1 h=2kT h=kT n h=2kT 1 Z.T / e e e D D 1 e h=kT I n 0 XD i.e., 1 Z.T / : (6.21) h D 2 sinh 2kT 56 FLOYDL.WILLIAMS

The importance of the partition function Z.T / is that from it one derives basic thermodynamic quantities such as def the Hemholtz free energy F.T / kT log Z.T /; defD the entropy S.T / @F=@T ; defD @F the internal energy U.T / F.T / TS.T / F.T / T : D C D @T Using (6.21), one computes that by these deﬁnitions h h F.T / kT log 1 exp ; D 2 C kT h=kT h S.T / k log 1 exp ; (6.22) D exp.h=kT / 1 kT h h U.T / ; D 2 C exp.h=kT / 1 which means that the factor h=.eh=kT 1/ of f. T / in equation (6.18) I 1 differs from the internal energy U.T / exactly by the quantity E0 h, which D 2 is the ground state energy (also called the zero-point energy) of the quantized 1 harmonic oscillator since En .n /h . However, our main interest is in D C 2 setting up a free energy – zeta function connection. Here’s how it goes. def For convenience let ˇ 1=.kT / denote the inverse temperature. Form the ﬁnite temperature zeta functionD

def 1 .s T / ; (6.23) I D .42n2 h22ˇ2/s n ޚ X2 C 1 which turns out to be well-defined and holomorphic for Re s > 2 . In [42] we show that .s T / has a meromorphic continuation to Re s < 1 given by I 1 2 s .s 2 / 1 2a sin s 1 .x 1/ .s T / 2.pa/ dx (6.24) I D p4.s/as 1=2 C 1 exp.pax/ 1 Z def for a h22ˇ2. Moreover .s T / is holomorphic at s 0 and D I D .0 T / pa 2 log 1 exp. pa/ (6.25) 0 I D I see Theorem 14.4 and Corollary 14.2 of [42]. By definition, pa hˇ D D h=kT . Therefore by formula (6.25) (which does require some work to derive from (6.24)), and by the first formula in (6.22), one discovers that kT F.T / .0 T /; (6.26) D 2 0 I which is the free energy – zeta function connection. LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 57

1 We will meet the zero-point energy E0 h again in the next lecture re- D 2 garding the discussion of Casimir energy. In chapter 16 of [42] another ﬁnite temperature zeta function is set up in the context of Kaluza–Klein space-times with spatial sector ޒm G=K, where is a discrete group of isometries of n the rank 1 symmetric space G=K; here K is a maximal compact subgroup of the semisimple Lie group G. In this broad context a partition function Z.T / and free energy-zeta function connection still exist.

Lecture 7. Zeta regularization, spectral zeta functions, Eisenstein series, and Casimir energy

Zeta regularization is a powerful, elegant procedure that allows one to assign to a manifestly infinite quantitiy a finite value by providing it a special value zeta interpretation. Such a procedure is therefore of enormous importance in physics, for example, where infinities are prolific. As a simple example, we consider the sum S 1 2 3 4 1 n, which is obviously infinite. This D C C C C D n 1 sum arises naturally in string theory —D in the discussion of transverse Virasoro operators, for example. A string P(which replaces the notion of a particle in 33 quantum theory, at the Plancktian scale 10 cm) sweeps out a surface called a world-sheet as it moves in d-dimensional space-time ޒd ޒ1 ޒd 1 - in D contrast to a world-line of a point-particle. For Bosonic string theory (where there are no fermions, but only bosons) certain Virasoro constraints force d to assume a specific value. Namely, the condition d 2 1 S (7.1) D 2 arises, which as we shall see forces the critical dimension d 26, 1 being the D s value of a certion normal ordering constant. In fact, we write S 1 1=n , D n 1 where s 1, which means that it is natural to reinterpret S as the specialD zeta D P value . 1/. Thus we zeta regularize the infinite quantity S by assigning to it the value 1 , according to (2.14). Then by condition (7.1), indeed we must 12 have d 26. D Interestingly enough, the “strange” equation 1 2 3 4 1 ; (7.2) C C C CD 12 which we now understand to be perfectly meaningful, appears in a paper of Ramanujan — though he had no knowledge of the zeta function. It was initially dismissed, of course, as ridiculous and meaningless. As another simple example we consider “ ! ”, that is, the product P 1 D 1 2 3 4 , which is also infinite. To zeta regularize P, we consider first log P n1 1 log n (which is still infinite), and we note that since 0.s/ D D D P 58 FLOYDL.WILLIAMS

s n1 1.log n/n for Re s > 1, by equation (1.1), if we illegally take s 0 the D D false result 0.0/ n1 1 log n log P follows. However the left-hand side P D D D 1 here is well-defined and in fact it has the value 2 log 2 by equation (2.6). The 1 P finite value 2 log 2, therefore, is naturally assigned to log P and, consequently, we define def 0 1 1 .0/ 2 log 2 P n ! e e p2: (7.3) D n 1 D 1 D D D D More complicated productsQ can be regularized in a somewhat similar manner. A typical set-up for this is as follows. One has a compact smooth manifold M with a Riemannian metric g, and therefore a corresponding Laplace–Beltrami operator .g/ where has a discrete spectrum D 0 0 < 1 < 2 < 3 < ; lim j : (7.4) D j D 1 !1 If nj denotes the (finite) multiplicity of the j-th eigenvalue j of , then one can form the corresponding spectral zeta function (cf. definition (0.1))

1 nj M .s/ ; (7.5) D s j 1 j XD 1 which is well-defined for Re s > 2 dim M , due to the discovery by H. Weyl 2=dim M of the asymptotic result j j , as j . S. Minakshisundaram and ! 1 A. Pleijel [26] showed that M .s/ admits a meromorphic continuation to the complex plane and that, in particular, M .s/ is holomorphic at s 0. Thus M .0/ D e is well-defined and, as in definition (7.3), we set

1 n def 0 det0 j e M .0/; (7.6) D j D j 1 YD where the prime 0 here indicates that the product of eigenvalues (which in finite dimensions corresponds to the determinant of an operator) is taken over the nonzero ones. Indeed, similar to the preceding example with the infinite product P j1 1 j , the formal, illegal computation D D Q d 1 nj 1 nj 1 exp exp log j exp nj log j ds s D s D j 1 j ˇs 0 j 1 j ˇs 0 j 1 XD ˇ D XD ˇ D XD ˇ ˇ ˇ 1 1 ˇ nj enj log j (7.7) D D j j 1 j 1 YD YD serves as the motivation for definition (7.6). Clearly this definition of determinant makes sense for more general operators (with a discrete spectrum) on other LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 59 infinite-dimensional spaces. It is useful, moreover, for Laplace-type operators on smooth sections of a vector bundle over M . The following example is more involved, where M is a complex torus (in fact M is assumed to be the world-sheet of a bosonic string; see Appendix C of [42], for example). For a fixed complex number 1 i2 in the upper D C half-plane (2 > 0) and for the corresponding integral lattice, def def L a b a; b ޚ ; M ރ L : (7.8) Df C j 2 g D n In this case it is known that has a multiplicity free spectrum (i.e., every nj 1) given by D 2 def 4 2 mn m n ; D 2 j C j m;n ޚ n 2 o 2 and consequently the corresponding spectral zeta function of (7.5) is given by s 2 M .s/ E .s;/ (7.9) D .42/s for Re s > 1, where s def E .s;/ 2 (7.10) 2s D 2 m n .m;n/ ޚ j C j X2 (with ޚ2 ޚ ޚ .0; 0/ as before) is a standard nonholomorphic Eisenstein D f g series. That is, in contrast to the series Gk./ in definition (4.4), E.s;/ is not a holomorphic function of . As a function of s, it is a standard fact that E.s;/, which is holomorphic for Re s > 1, admits a meromorphic continuation to the full complex plane, with a simple pole at s 1 as its only singularity. Hence, D by equation (7.9), the same assertion holds for M .s/. By [7; 14; 35; 38], for example, the continuation of E.s;/ is given by .s 1 / E .s;/ 2.2s/ s 2.2s 1/p 2 s 1 D 2 C .s/ 2 C s 1 4 1 1 n s 1=2 2imn1 2 e K 1 .2mn2/ C .s/ 2 m s m 1 n 1 2 XD XD s 1 4 1 1 n s 1=2 2imn1 2 e K 1 .2mn2/ (7.11) C .s/ 2 m s m 1 n 1 2 XD XD for Re s > 1, where

def 1 1 x 1 1 K.x/ exp t t dt (7.12) D 2 0 2 C t Z 60 FLOYDL.WILLIAMS is the Macdonald–Bessel (or K-Bessel) function for ރ, x > 0. Introduc- def 2 ing the divisor function .n/ d , and using forumla (D.11) on D 0

.s 1 / 2.2s/ s 2.2s 1/p 2 s 1 D 2 C .s/ 2 C s 1 4 1=2 1 2n1i s 2 2 2s 1.n/e K 1 .2n2/n C .s/ C s n 1 2 XD s 1 4 1=2 1 2n1i s 2 2 2s 1.n/e K 1 .2n2/n : (7.13) C .s/ C s n 1 2 XD

The sum of the ﬁrst two terms in (7.13) has s 1 as a removable singularity. D 2 To see this, note ﬁrst that since .s/ has residue 1 at s 1, D D

1 s 1 z=2 lim s .2s/2 lim .z 1/.z/ 1 2 D 2 z 1 2 s ! ! 2

(for z 2s), and this equals 1=2=2. We also have .1/ 1, . 1 / p, D 2 D 2 D .0/ 1 (by (2.5)), and w .w/ .w 1/. Thus D 2 D C z 1 z 1 z 1 C 2 2 D 2 I hence

1 1 .s 2 / s 1 lim s .2s 1/p C 1 2 .s/ 2 s z 1 ! 2 z 1 z=2 1 lim .z 1/p 2 z 2 C D z 1 2 ! 2 p z 1 1=2 1=2 .0/ lim C =2: D . 1 / z 1 2 2 D 2 2 !

It follows that the limit as s 1 of .s 1 / the first two terms in (7.13) ! 2 2 vanishes, as desired. This proves our claim that s 1 is the only singularity D of E .s;/, which arises as a simple pole from the second term, 2.2s 1/ p.s 1 /= .s/ s 1, in (7.13), due to the factor .2s 1/. Moreover the 2 2 C LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 61 residue at s 1 can be easily evaluated setting z 2s 1: D D 1 s 1 lim .s 1/2.2s 1/p.s /= .s/ C s 1 2 2 ! z z 1 z=2 1=2 lim .z 1/.z/p C C D z 1 2 2 2 ! p.p=1/ : ı D D s From (7.13) we also get E.0;/ 2.2s/2 s 0 D D W E .0;/ 1ˇ: (7.14) D ˇ Moreover, the functional equation E .1 s;/ 1 2sE .s;/ ; (7.15) .s/ D .1 s/ say for s 0; 1, follows since .n/, K.x/ satisfy the functional equations ¤ .n/ n .n/; K.x/ K .x/; (7.16) D D and since .s/ satisfies the functional equation (1.16). The second equation in (7.16) follows by the change of variables u 1=t in definition (7.12), and the D first equation is the formula

1 .n/ n ; D d 0 0 runs through the divisors of n as n does. We check equation (7.15) by d 1 1 s 2 replacing s by 1 s in equation (7.13). By (7.16), 2.1 s/ 1.n/n s 1 C D 2s 1n 2 and K 1 .2n2/ K 1 .2n2/; hence 1 s 2 s 2 C D 1 s 1 4 1=2 1 2n1i 1 s 2 2 2.1 s/ 1.n/e˙ K 1 .2n2/n .1 s/ C 1 s D n 1 2 XD 1 s 1 4 1=2 1 2n1i s 2 2 2s 1.n/e˙ K 1 .2n2/n : (7.17) .1 s/ C s n 1 2 XD In equation (1.16) replace s by 2s and 2s 1 separately, to obtain 1 2s C 2 .s/.2s/ .1 2s/ ; D . 1 s/ 2 .s 1 /.2s 1/ .2 2s/ .1 .2s 1// 1=21 2s 2 D D .1 s/ 62 FLOYDL.WILLIAMS say for 2s; 2s 1 0; 1 (i.e., s 0; 1 ; 1). This gives ¤ D 2 1 1 s .1 s 2 / .1 s/ 1 2.2.1 s// 2.2.1 s/ 1/p C 2 C .1 s/ 2 .s 1 /.2s 1/ 2p1 2s 2 s 1 D .1 s/ 2 C 1 2s 1 2 C 2 .s/.2s/p. s/ 2 s: (7.18) C . 1 s/.1 s/ 2 2 E .1 s;/ By (7.13), (7.17), (7.18), we have for the value .s/ 1 2s .s 1 / 2p 2 .2s 1/ s 1 2.2s/ s .1 s/ .s/ 2 C C 2 1 2s s 1 4 1=2 1 2n1i s 2 2 2s 1.n/e K 1 .2n2/n C .1 s/ .s/ C s n 1 2 D s 1 4 P1=2 1 2n1i s 2 2 2s 1.n/e K 1 .2n2/n C .s/ C s n 1 2 D 1 2s P E .s;/ D .1 s/ which gives (7.15), as desired. (Note that we have stayed away from s 0; 1 ; 1; D 2 but equation (7.15) clearly holds for s 1 .) D 2 One other result is needed in order to compute M0 .0/: THEOREM 7.19 (KRONECKER’S FIRST LIMIT FORMULA).

1=2 2 lim E.s;/ 2 log 2 log ./ : (7.20) s 1 s 1 D 2 j j ! where is the Euler–Mascheroni constant in definition (1.26), and where ./ is the Dedekind eta function in definition (3.27). Formula (7.20) compares with the limit result (1.25), though it is a more involved result; see [35]. def For f.z/ .z/2 2z= .2 z/ D f .1/ 2 log 2 .1/ 2 log 2 (7.21) 0 D C 0 D by the quotient rule. By equations (7.14), (7.15), (7.21) and Theorem 7.19, and the fact that .1 z/.1 z/ .2 z/ (i.e., w .w/ .w 1/), we have, D D C LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 63 with z 1 s, D @E def E .s;/ 1 E .1 z;/ 1 .0;/ lim C lim C @s D s 0 s D z 1 1 z ! ! E .1 z;/.z/ 1 lim Dz 1 .z/.1 z/ z 1 ! .z/ 1 2z E.z;/ 1 lim Dz 1 .1 z/ .1 z/ z 1 ! 1 2z 1 2z .z/ .z/ 1 lim E.z;/ Dz 1 .2 z/ z 1 C .2 z/ z 1 z 1 ! def 1=2 2 f.z/ f.1/ 2Œ log 2 log ./ lim f 0.1/ D 2 j j C z 1 z 1 D ! 2 log 2 2 log 1=2 ./ 2 2 log D 2 j j 2 4 log 4 2 ./ ; D j j which, with equations (7.9), (7.14) gives (finally)

.0/ log 2 ./ 4 : (7.22) M0 D 2 j j Hence, by deﬁnition (7.6), the regularized determinant is given by

det0 2 ./ 4 : (7.23) D 2 j j d=2 d 2d One is actually interested in the power .det / ./ , where 0 D 2 j j d 26 is the critical dimension mentioned above. This power represents a one- D loop contribution to the “sum of embeddings” of the string world sheet (the complex torus in deﬁnition (7.8)) into the target space ޒ26. In the next lecture we shall make use, similarly, of the meromorphic continuation of the generalized Epstein zeta function

def 2 2 2 s E.s; m a; bE/ a1.n1 b1/ ad .nd bd / m (7.24) IE D d CC C n .n1;n2;:::;nd / ޚ ED X 2 d for Re s > d=2, m > 0, a .a1;:::; a /, b .b1;:::; b / ޒ , ai > 0. The E D d E D d 2 result is, setting ޚd ޚd 0 , D f g d=2.s d /md 2s 2smd=2 s E.s; m a; b/ 2 IE E D pa1a2 a .s/ C pa1a2 a .s/ d d d 2 s d=2 d 2 1 d nj 2 nj 2 e2i j D1 nj bj K 2m (7.25) d s d P j 1 aj 2 j 1 aj n ޚ D D EX2 P P 64 FLOYDL.WILLIAMS for Re s > d=2. In particular, we shall need the special value E. d 1 ; m a; b/, C2 IE E which we now compute. For s d 1 the ﬁrst term on the right in (7.25) is D C2 d 1 C2 ; d 1=2 d 1 m j 1 aj C2 D since 1 1=2: Also Q 2 D K .x/ K .x/ K .x/; d s 1 1 2 D 2 D 2 x by (7.16). This equals =2x e for x > 0; hence 2 s d=2 2 1 d n p d n j 2 K 2m j 2 d s j 1 aj 2 j 1 aj D D P 2 P 2 1 2 1 d n 1=4 1 d n d n j j 4 j 2 .2m/ 2 exp 2m : D a 2 a a j 1 j r j 1 j j 1 j PD PD PD Therefore the second term on the right-hand side of (7.25) is

d 1 2 1 2 C2 m 1=2 1 d d n exp 2i n b exp 2m j 2 : 1=2 j j d d 1 2pm aj aj C d j 1 j 1 j 1 2 n ޚ D D D EX2 P P ThatQ is: d 1 d 1 C2 E C ; m a; b 2 IE E D d 1=2 d 1 m j 1 aj C2 D d 1 2 1 C2 Q d n 2in b j 2 e E exp 2m 1=2 E C d d 1 aj m aj C d j 1 j 1 2 n ޚ D D EX2 P d 1 2 1 Q C2 d n 2in b j 2 e EE exp 2m : (7.26) D d 1=2 d 1 a m aj d j 1 j j 1 C2 n ޚ D D EX2 P As a finalQ example we consider the zeta regularization of Casimir energy, after a few general remarks. def 1 In Lecture 6 it was observed that the sequence En n h is the D C 2 sequence of energy levels of the quantized harmonic oscillator of frequency , ˚ « where h denotes Planck’s constant. In particular there exists a nonvanishing 1 ground state energy (also called the zero-point energy) given by E0 h. D 2 Zero-point energy is a prevalent notion in physics, from quantum field theory (QFT), where it is also referred to as vacuum energy, to cosmology (concerning LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 65 issues regarding the cosmological constant, for example), and in between. Based on Planck’s radiation density formula (6.18), A. Einstein and O. Stern concluded (in 1913) that even at zero absolute temperature, atomic systems maintain an 1 energy of the amount E0 h. It is quite well experimentally established D 2 that a vacuum (empty space) contains a large supply of residual energy (zero- point energy). Vacuum fluctuations is a large scale study. The energy due to vacuum distortion (Casimir energy), for example, was considered by H. Casimir and D. Polder in a 1948 ground-breaking study. Here the vacuum energy was modified by the introduction of a pair of uncharged, parallel, conducting metal plates. A striking prediction emerged: the prediction of the existence of a force of a purely quantum mechanical origin — one arising from zero-point energy changes of harmonic oscillators that make up the normal modes of the electromagnetic field. This force, which has now been measured experimentally by M. Spaarnay, S. Lamoreaux, and others, is called the Casimir force. Casimir energy in various contexts has been computed by many Physicists, including some notable calculations by the co-editor Klaus Kirsten. We refer to his book [22] for much more information on this, and on related matters - a book with 424 references. The author has used the Selberg trace formula for general compact space forms G=K, mentioned in Lecture 6, of rank-one symmetric n spaces G=K to compute the Casimir energy in terms of the Selberg zeta function [40; 39]. This was done by Kirsten and others in some special cases. Consider again a compact smooth Riemannian manifold .M; g/ with discrete spectrum of its Laplacian .g/ given by (7.4). In practice, M is the spatial sector of a space-time manifold ޒ M with metric dt 2 g. Formally, the C Casimir energy in this context is given by the infinite quantity

1 1 1=2 EC nj ; (7.27) D 2 j j 1 XD up to some omitted factors like h. It is quite clear then how to regularize EC . 1=2 s 1 Namely, consider as 1= for s and therefore assign to EC the j j D 2 meaning 1 EC M . 1=2/; (7.28) D 2 where M .s/ is the spectral zeta function of definition (7.5), meromorphically continued. If dim M is even, for example, the poles of M .s/ are simple, finite in number, and can occur only at one of points s 1; 2; 3;:::; d=2 (see [26]), D in which case EC in (7.28) is surely a well-defined, finite quantity. However if dim M is odd, M .s/ will generally have infinitely many simple poles — at the points s 1 dim M n, for 0 n ޚ. This would include the point s 1 D 2 2 D 2 if dim M 5 and n 3, for example. Assume therefore that dim M is even. D D 66 FLOYDL.WILLIAMS

When M is one of the above compact space forms, for example, then (based on the results in [41]) EC can be expressed explicitly in terms of the structure of and the spherical harmonic analysis of G=K — and in terms of the Selberg zeta function attached to G=K, as just mentioned. Details of this are a bit too n technical to mention here; we have already listed some references. We point out only that by our assumptions on M , the corresponding Lie group pairs .G; K/ are given by

G SO1.m; 1/; K SO.m/; m 2; D D G SU.m; 1/; K U.m/; m 2; D D G SP.m; 1/; K SP.m/ SP.1/; m 2; D D G F4. 20/; K Spin.9/; D D where F4. 20/ is a real form of the complex Lie group with exceptional Lie algebra F4 with Dynkin diagram 0—0 0—0. More speciﬁcally, F4. 20/ is H the unique real form for which the difference dim G=K dim K assumes the value 20. In addition to the reference [22], the books [13; 12] are a good source for information on and examples of Casimir energy, and for applications in general of zeta regularization.

Lecture 8. Epstein zeta meets gravity in extra dimensions

We compute the Kaluza–Klein modes of the 4-dimensional gravitational potential V4 d in the presence of d extra dimensions compactified on a d-torus. The resultC is known of course [3; 21], but we present here an argument based on the special value E d 1 ; m a; b computed in equation (7.26) of the gener- C2 IE E alized Epstein zeta function E.s; m a; b/ defined in (7.24). IE E G. Nordstrom¨ in 1914 and T. Kaluza (independently) in 1921 were the first to unify Einstein’s 4-dimensional theory of gravity with Maxwell’s theory of electromagnetism. They showed that 5-dimensional general relativity contained both theories, but under an assumption that was somewhat artificial - the so- called “cylinder condition” that in essence restricted physicality of the fifth dimension. O. Klein’s idea was to compactify that dimension and thus to render a plausible physical basis for the cylinder assumption. Consider, for example, the fifth dimension (the “extra dimension”) compactified on a circle . This means that instead of considering the Einstein gravitational field equations

gij 8G Rij .g/ R.g/ gij Tij (8.1) 2 D c4 LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 67 on a 4-dimensional space-time M 4 [8; 11], one considers these equations on 4 the 5-dimensional product M . In (8.1), g Œgij is a Riemannian metric D (the solution of the Einstein equations) with Ricci tensor Rij .g/ and scalar curvature R.g/; is a cosmological constant, and Tij is an energy momentum tensor which describes the matter content of space-time — the left-hand side of (8.1) being pure geometry; G is the Newton constant and c is the speed of light. Given the non-observability of the ﬁfth dimension, however, one takes to be extremely small, say with an extremely small radius R > 0. Geometrically we have a ﬁber bundle M 4 M 4 with structure group . !

M 4

On all “ﬁelds” F.x;/ M 4 ޒ ރ on M 4 ޒ there is imposed, moreover, W ! periodicity in the second variable:

F.x; 2R/ F.x;/ (8.2) C D for .x;/ M 4 ޒ. 2 def For n ޚ and f.x/ on M 4 fixed, the function F .x;/ f.x/ein=R is 2 n;f D an example of a field on M 4 ޒ that satisfies equation (8.2). For a general field F.x;/, subject to reasonable conditions, and the periodicity condition (8.2), one would have a Fourier series expansion

def in=R F.x;/ F fn.x/e (8.3) D n;fn D n ޚ n ޚ X2 X2 in which case the functions fn.x/ are called Kaluza–Klein modes of F.x;/. Next we consider d extra dimensions compactiﬁed on a d-torus

d def 1 ; D d where the j are circles with extremely small radii Rj > 0, and we consider a 3 d 3 4 ﬁeld V4 d .x; y; z; x1;:::; xd / on .ޒ 0 / ޒ . Thus ޒ 0 replaces M , C f g f g 68 FLOYDL.WILLIAMS

d def d replaces , and x .x1;:::; x / ޒ replaces in the previous discus- E D d 2 sion. The ﬁeld is given by def V4 d .x; y; z; x1;:::; xd / C D 1 MG4 d ; (8.4) d d 1 C d C n .n1;:::;nd / ޚ 2 2 2 ED X 2 r .xj 2nj Rj / Cj 1 PD which is the gravitational potential due to extra dimensions of an object of mass 2 d 2 1=2 2 def 2 2 2 M at a distance r j 1 xj for r x y z ; here G4 d is the C D C C C .4 d/-dimensional NewtonD constant. Note that, analogously to equation (8.2), C P V4 d .x; y; z; x1 2R1;:::; xd 2Rd / C C C def 1 MG4 d d d 1 D C d C .n1;:::;nd / ޚ 2 2 2 X 2 r .xj 2.n 1/jRj / Cj 1 PD 1 (8.5) MG4 d d d 1 D C d C .n1;:::;nd / ޚ 2 2 2 X 2 r .xj 2nj Rj / Cj 1 D V4 d .x; y; z; x1;:::; xd /; P D C where of course we have used that nj 1 varies over ޚ as nj does. Thus, analogously to equation (8.3), we look for a Fourier series expansion

!x1 xd V4 d .x;y;z;x1;:::;xd / fn.x;y;z/exp in ;:::; ; (8.6) C D E E R1 Rd n ޚd EX2 ޒ3 where the functions fn.x; y; z/ on 0 would be called the Kaluza–Klein E f g modes of V4 d .x; y; z; x .x1;:::; xd //. C E D It is easy, in fact, to establish the expansion (8.6) and to compute the modes fn.x; y; z/ explicitly. For this, define E def 2 def xj aj .2Rj / > 0; bj ; D D 2Rj 2 2 xj 2 and note that since .xj 2nj Rj / 2Rj nj aj .bj nj / D 2Rj D we can write, by definition (8.4), 1 V4 d .x; y; z; x/ MG4 d C E D C d d 1 n ޚd 2 2 C2 EX2 aj .nj bj / r j 1 C D d 1P MG4 d E C ; r a; b ; (8.7) D C 2 IE E LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 69 by definition (7.24). Thus we are in a pleasant position to apply formula (7.26): For d d 1 C2 def d def 2 ˙ .2/ Rj ; ˝ (8.8) d D d D d 1 j 1 C YD 2 d 1=2 we have j 1 aj ˙d , and we see that D D V4 d .x;yQ; z; x/ C E D d 2 1 MG4 d ˝d x x n 2 exp in !1 ;:::; d exp r j (8.9) C R R 2 2r˙d d E 1 d j 1 Rj n ޚ D EX2 P 2 def 2 2 2 by definition of aj and bj , for r x y z . This proves the Fourier series D C C expansion (8.6), where we see that the Kaluza–Klein modes fn.x; y; z/ are in fact given by E

d 2 1 MG4 d ˝d nj 2 f .x; y; z/ C exp r (8.10) n D 2r˙ R2 E d j 1 j PD d 3 for n .n1;:::; nd / ޚ ;.x; y; z/ ޒ 0 . Since V4 d .x; y; z; x/ is E D 2 2 f g C E actually real-valued, we write equation (8.9) as

V4 d .x; y; z; x1;:::; xd / C D d 2 1 MG4 d ˝d n 2 x x exp r j cos n !1 ;:::; d : (8.11) C 2 R R 2r˙d d j 1 Rj E 1 d n ޚ D EX2 P d def d d Since 2Ri is the length of i; i 1 i has volume i 1 2Ri d d D D D D .2/ Ri. That is, ˙ in deﬁnition (8.8) (or in formula (8.11)) is the i 1 d Q Q volume ofD the compactifying d-torus d . Similarly ˝ in (8.8) or in (8.11), Q d one knows, is the surface area of the unit sphere x ޒd 1 x 1 in ޒd 1. 2 C jk kD C In [21], for example, the choice x1 x 0 is made. Going back to DD ˚d D « the compactiﬁcation on a circle, d 1; R1 R for example, we can write the D D sum in (8.11) as

1 n 1 e r n =R 1 2 e r=R C j j D C n ޚ 0 n 1 2Xf g XD r=R 2e r=R 1 1 2e (8.12) D C 1 e r=R ' C for x1 0, where we keep in mind that R is extremely small. Thus in (8.12), D r=R r=R is extremely large; i.e., e is extremely small. For def Kd MG4 d ˝d =2˙d ; D C 70 FLOYDL.WILLIAMS we get by (8.11) and (8.12)

K1 r=R V5.x; y; z; x1/ .1 2e /; (8.13) ' r C which is a correction to the Newtonian potential V K1=r due to an extra D dimension. The approximation (8.13) compares with the general deviations from the Newtonian inverse square law that are known to assume the form K V .1 ˛e r=/ D r C for suitable parameters ˛; . Apart from the toroidal compactification that we have considered, other compactifications are important as well [21] — especially Calabi–Yau compactifications. Thus the d-torus d is replaced by a Calabi– Yau manifold — a compact Kahler¨ manifold whose first Chern class is zero.

Lecture 9. Modular forms of nonpositive weight, the entropy of a zero weight form, and an abstract Cardy formula

A famous formula of John Cardy [9] computes the asymptotic density of states .L0/ (the number of states at level L0) for a general two-dimensional conformal ﬁeld theory (CFT): For the holomorphic sector

2pcL0=6 .L0/ e ; (9.1) D where the Hilbert space of the theory carries a representation of the Virasoro algebra Vir with generators Ln n ޚ and central charge c. Vir has Lie algebra f g 2 structure given by the usual commutation rule

c 2 ŒLn; Lm .n m/Ln m n.n 1/ın m;0 (9.2) D C C 12 C for n; m ޚ. The CFT entropy S is given by 2 cL0 S log .H0/ 2 : (9.3) D D r 6 From the Cardy formula one can derive, for example, the Bekenstein–Hawking formula for BTZ black hole entropy [10]; see also my Speaker’s Lecture presented later. More generally, the entropy of black holes in string theory can be derived — the derivation being statistical in nature, and microscopically based [34]. For a CFT on the two-torus with partition function c Z./ trace e2i.L0 24 / (9.4) D LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 71 on the upper half-plane C [5], the entropy S can be obtained as follows. Re- garding Z./ as a modular form with Fourier expansion

2i.n c=24/ Z./ cne ; (9.5) D n 0 X one takes S log cn (9.6) D for large n. In [5], for example, (also see [4]) the Rademacher–Zuckerman exact formula for cn is applied, where Z./ is assumed to be modular of weight w 0. D This is problematic however since the proof of that exact formula works only for modular forms of negative weight. In this lecture we indicate how to resolve this contradiction (thanks to some nice work of N. Brisebarre and G. Philibert), and we present what we call an abstract Cardy formula (with logarithmic correction) for holomorphic modular forms of zero weight. In particular we formulate, ab- stractly, the sub-leading corrections to Bekenstein–Hawking entropy that appear in formula (14) of [5]. The discussion in Lecture 4 was confined to holomorphic modular forms of non-negative integral weight. We consider now forms of negative weight w r for r > 0, where r need not be an integer. The prototypic example D def will be the function F0.z/ 1=.z/, where .z/ is the Dedekind eta function D defined in (3.27), and where it will turn out that w 1 . We will use, in fact, D 2 the basic properties of F0.z/ to serve as motivation for the general definition of a form of negative weight. We begin with the partition function p.n/ on ޚC. For n a positive integer, define p.n/ as the number of ways of writing n as an (orderless) sum of positive integers. For example, 3 is expressible as 3 1 2 1 1 1, so p.3/ 3; D C D C C D 4 4 1 3 2 2 1 1 2 1 1 1 1, so p.4/ 5; D D C D C D C C D C C C D 5 5 2 3 1 4 1 1 3 1 2 2 1 1 1 2 1 1 1 1 1; D D C D C D C C D C C D C C C D C C C C def so p.5/ 7; similarly p.2/ 2, p.1/ 1. We set p.0/ 1. Clearly p.n/ D D D D grows quite quickly with n. A precise asymptotic formula for p.n/ was found by G. Hardy and S. Ramanujan in 1918, and independently by J. Uspensky in 1920: ep2n=3 p.n/ as n (9.7) 4np3 ! 1 (the notation means that the ratio between the two sides of the relation (9.7) tends to 1 as n ). For example, it is known that ! 1 p.1000/ 24;061;467;864;032;622;473;692;149;727;991 D 2:4061 1031; (9.8) ' 72 FLOYDL.WILLIAMS whereas for n 1000 in (9.7) D 2n e 3 =4np3 2:4402 1031; (9.9) q ' which shows that the asymptotic formula is quite good. L. Euler found the generating function for p.n/. Namely, he showed that

1 1 p.n/zn (9.10) n D 1 .1 z / n 0 n 1 XD D for z ރ with z < 1. By this formulaQ and the deﬁnition of .z/, we see 2 j j immediately that

def 1 i=12 1 2in F0.z/ e p.n/e (9.11) D .z/ D n 0 XD on C. The following profound result is due to R. Dedekind. To prepare the ground, for x ޒ define 1 2 def x Œx if x ޚ, ..x// 2 62 D 0 if x ޚ, 2 where, as before, Œx denotes the largest integer not exceeding x. a b def THEOREM 9.12. Fix SL.2; ޚ/, with c > 0, and define D c d 2 D a d 1 S. / C s.d; c/; D 12c 4 where s.d; c/ (called a Dedekind sum) is given by def d s.d; c/ : (9.13) D c c ޚ=cޚ 2X Then, for z , 2 C 1 1 i(S. / ) 2 F0. z/ e C 4 i.cz d/ F0.z/ (9.14) D C for =2 < arg i.cz d/ < =2, where z is defined in equation (4.3). C The sum in definition (9.13) is over a complete set of coset representatives in ޚ. The case c 0 is much less profound; then D 1 b ˙ D 0 1 ˙ (since 1 det ad), and D D ib=12 F0.z b/ F0. z/ e F0.z/: (9.15) ˙ D D LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 73

i=12 2i=24 2i In particular we can write F0.z 1/ e F0.z/ e C F0.z/ 2i.1 1 / 2i˛ C Ddef 1 23 D D e 24 F0.z/ e F0.z/ for ˛ 1 . D D 24 D 24 In summary, F0.z/ 1=.z/ satisfies the following conditions: D (i) F0.z/ is holomorphic on C. (This follows from Lecture 3).) 2i˛ 23 (ii) F0.z 1/ e F0.z/ for some real ˛ Œ0; 1/ (indeed, with ˛ 24 ). C D r2 a b D (iii) F0. z/ ".a; b; c; d/ i.cz d/ F0.z/ for with D C D c d 2 c > 0, for some r > 0, =2 < arg i.cz d/ < =2, and for a function C ". / ".a; b; c; d/ on with ". / 1 (indeed, for r 1 and ".a; b; c; d/ 2 D a d j jD D D exp i 12cC s.d; c/ , by Theorem 9.12). 2i˛z 2inz (iv) F0.z/ e n1 ane on C for some integer 1 (indeed, D D for 1, an p.n 1/ for n 1, and an 0 for n 2, by Euler’s D D C D formula (9.11)). P def 2i˛z Note that by conditions (i) and (ii), the function f.z/ e F0.z/ is holo- D morphic on C, and it satisfies f.z 1/ f.z/. Thus, again by equation C D 2inz (4.1), f.z/ has a Fourier expansion f.z/ n ޚ ane on C. That is, D 2 conditions (i) and (ii) imply that F0.z/ has a Fourier expansion P 2i˛z 2inz F0.z/ e ane D n ޚ P2 on C, and condition (iv) means that we require that a n 0 for n >, for D some positive integer . We abstract these properties of F0.z/ and, in general, we define a modular form of negative weight r, for r > 0, with multiplier " z ރ z 1 W !f 2 j j jD g to be a function F.z/ on C that satisfies conditions (i), (ii), (iii), and (iv) for some ˛ and with 0 ˛ < 1, ޚ, 1. Thus .z/ 1 is a modular form of 2 weight 1 and multiplier ".a; b; c; d/ exp i a d s.d; c/ , with ˛ 23 , 2 D 12cC D 24 1, and with Fourier coefficients an p.n 1/, as we note again. D D C For modular forms of positive integral weight, there are no general formulas available that explicitly compute their Fourier coefficients — apart from Theo- rem 4.23 for holomorphic Eisenstein series. For forms of negative weight however, there is a remarkable, explicit (but complicated) formula for their Fourier coefficients, due to H. Rademacher and H. Zuckerman [31]; also see [29; 30]. Before stating this formula we consider some of its ingredients. First, we have the modified Bessel function t 2m def t 1 2 I.t/ (9.16) D 2 m!. m 1/ m 0 XD C C for t > 0; ރ; the series here converges absolutely by the ratio test. Next, for 2 k; h ޚ with k 1; h 0;.h; k/ 1, and h < k choose a solution h of the 2 D 0 74 FLOYDL.WILLIAMS congruence hh 1.mod k/. For example, .h; k/ 1 means that the equation 0 D xh yk 1 has a solution .x; y/ ޚ ޚ. Then xh 1 yk means C def D 2 D C that h x is a solution. Since hh 1 lk for some l ޚ, we see that 0 D 0 D C 2 .hh 1/=k l is an integer and 0 C D h .hh 1/=k def h .hh 1/=k det 0 0C 1; so 0 0C : k h D D k h 2 Hence hh0 1 ". / " h0; C ; k; h (9.17) D k is well-defined. Finally, for u;v ރ we define the generalized Kloosterman sum 2

2i 0 def 1 ..u ˛/h .v ˛/h/ A .v/ A .v; u/ ". / e k C C (9.18) k;u D k D 0 h 0, with multiplier ", and with Fourier 2i˛z 2inz expansion F.z/ e 1 ane on given by condition (iv) above, D n C where 0 ˛ < 1 ޚ. ThenD for n 0 with not both n;˛ 0, 2 P D an D r 1 1 Ak;j .n/ j ˛ C2 4 1=2 1=2 2 a j Ir 1 .j ˛/ .n ˛/ ; (9.21) k n ˛ C k C j 1 k 1 C XD XD where Ak;j .n/ is deﬁned in (9.18) and I.t/ is the modiﬁed Bessel function in (9.16). Note that for 1 j ; j 1 >˛ j ˛ > 0 in equation (9.21). Also ) n ˛ > 0 there since n;˛ 0 with not both n;˛ 0. C D Using the asymptotic result

t lim p2tI.t/e 1 (9.22) t D !1 for the modiﬁed Bessel function I.t/ in (9.16), and also the trivial estimate A .n/ k that follows from (9.18), one can obtain from the explicit formula j k;j j LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 75

(9.21) the following asymptotic behavior of an as n . Assume that a 0 ! 1 ¤ and define r 1 2 4 def a . ˛/ C 1=2 1=2 a1.n/ "0 exp 4. ˛/ .n ˛/ ; (9.23) D p2 r 3 C .n ˛/ 2 C 4 C def 0 1 say for n 1, for "0 " in (9.19). Then in [23], for example, it is D 1 0 shown that an a .n/ as n (9.24) 1 ! 1 which gives the asymptotic behavior of the Fourier coefficients of a modular form F.z/ of negative weight r with Fourier expansion as in the statement of Theorem 9.20. For forms of zero weight a quite similar result is given in equation (9.30) below. The asymptotic result (9.7) follows from (9.24) applied to F0.z/, in which case formula (9.21) provides an exact formula (due to Rademacher) for p.n/ [2; 29; 30]. For a; b; k ޚ with k 1 the classical Kloosterman sum S.a; b k/ is defined 2 I by 2i k .ah bh/ S.a; b k/ e C N (9.25) I D h2ޚ=kޚ .hX;k/ 1 D where hh 1.mod k/. These sums will appear in the next theorem (Theorem N 9.27) that is a companion result of Theorem 9.20. We consider next modular forms F.z/ of weight zero. That is, F.z/ is a holomorphic function on such that F. z/ F.z/ for , and with C D 2 Fourier expansion

1 2inz F.z/ ane (9.26) D n XD on C, for some positive integer . In case F.z/ is the modular invariant j.z/, for example, this expansion is that given in equation (4.44) with 1, in which D case the an there are computed explicitly by H. Petersson and H. Rademacher [27; 28], independently - by a formula similar in structure to that given in (9.21). For the general case in equation (9.26) the following extension of the Petersson– Rademacher formula is available [6]:

THEOREM 9.27 (N. BRISEBARRE AND G. PHILIBERT). For a modular form F.z/ of weight zero with Fourier expansion given by equation (9.26), its n-th Fourier coefﬁcient an is given by j 1 S.n; j k/ 4pnj an 2 a j I I1 (9.28) D n k k j 1 r k 1 XD XD 76 FLOYDL.WILLIAMS for n 1, where S.n; j k/ is deﬁned in (9.25) and I1 .t > 0/ in (9.16). I M. Knopp’s asymptotic argument in [23] also works for a weight zero form (as he shows), provided the trivial estimate A .n/ k used above is replaced j k;j j by the less trivial Weil estimate S.a; b k/ C."/.a; b; k/1=2k1=2 "; " > 0. j I j C 8 The conclusion is that if a 0, and if ¤ 1=4 def a 4pn a1.n/ e ; n 1; (9.29) D p2 n3=4 then an an a1.n/ lim 1: (9.30) W n a .n/ D !1 1

We see that, formally, deﬁnition (9.29) is obtained by taking "0 1; r 0; D D and ˛ 0 in deﬁnition (9.23) - in which case formulas (9.21) and (9.28) are D also formally the same. Going back to the Fourier expansion of the modular invariant j.z/ given in equation (4.44), where a a 1 1, we obtain from D D (9.30) that ([27; 28]) e4pn an as n : (9.31) p2n3=4 ! 1 A stronger result than (9.31), namely that

e4pn 3 :055 an 1 "n ; "n (9.32) D p 3=4 32pn C j j n 2n (also due to Brisebarre and Philibert [6]) plays a key role in my study of the asymptotics of the Fourier coefﬁcients of extremal partition functions of certain conformal ﬁeld theories; see Theorem 5-16 of my Speaker’s Lecture (page 345), and the remark that follows it. Motivated by physical considerations, and by equation (9.5) in particular, we consider a modular form of weight zero with Fourier expansion

2iz 2inz f.z/ e cne (9.33) D n 0 X on , where we assume that is a negative integer. corresponds to c=24 C in (9.5), say for a positive central charge c; thus c 24. /, a case considered def D def in my Speaker’s Lecture. is a positive integer such that for an cn , D D C we have (taking cn 0 for n 1) a n 0 for n > . Moreover, since D D LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 77

def 2inz n1 0 dn n1 dn, we see that for dn ane we have D D D D P P 1 2inz 1 2i.n /z ane an e n D n 0 DP PD def 1 2i.n /z cne C f.z/; (9.34) D n 0 D PD by (9.33). That is, f.z/ has the form (9.26), which means that we can apply formula (9.28), and the asymptotic result (9.30). Assume that c0 0, and define ¤ 1=4 def c0 4 1=2.n /1=2 c1.n/ j j e j j C (9.35) D p2 .n /3=4 C def for n 1. By definition (9.29), for n n 1, C D C 1=4 a 4p.n / def a1.n / e c1.n/; D p2.n /3=4 D def as a c0 0. Therefore by (9.30) D ¤ an cn 1 lim lim cn c1.n/ as n ; (9.36) D n a .n / D n c .n/ W ! 1 !1 1 !1 1 for c1.n/ defined in (9.35). Thus (9.36) gives the asymptotic behavior of the Fourier coefficients cn of the modular form f.z/ of weight zero in (9.33). Motivated by equation (9.6), and given the result (9.36) we define entropy function S.n/ associated to f.z/ by def S.n/ log c .n/ (9.37) D 1 for n 1, in case c0 > 0. Also we set C def S0.n/ 2 4 .n /; (9.38) D j j C for n 1. Then for c 24. /p 24 , as considered above, (i.e., for C D D j j 4 c=6) S0.n/ corresponds to the CFT entropy in equation (9.3), where j j D n corresponds to the L0 there. Moreover, by definition (9.37) we obtain C 1 3 1 S.n/ S0.n/ log log.n / log 2 log c0 ; (9.39) D C 4 j j 4 C 2 C which we can regard as an abstract Cardy formula with logarithmic correction, given by the four terms parenthesized. Note that, apart from, the term log c0, equation (9.39) bears an exact resemblance to equation (5.22) of my Speaker’s Lecture. We regard S0.n/ in definition (9.38), of course, as an abstract Bekenstein–Hawking function associated to the modular form f.z/ in (9.33) of zero weight. Equation (9.39) also corresponds to equation (14) of [5]. 78 FLOYDL.WILLIAMS

To close things out, we also apply formula (9.28) to f.z/. For n 1, j S.n ; j k/ 4 .n /j cn an 2 a j I I1 : (9.40) D D n k k j 1 r p XD 1 def Use j 1 dj d d 1 : : : d2 d1 j 0 d j and a . j/ cj D C C C C D D to writeD equation (9.40) as D P P 1 j S.n ; . j / k/ 4 .n /. j / cn 2 cj I I1 (9.41) D n k k j 0 r p XD def where . For 0 j 1 1; j ޚ; we have j 0 and D D 2 j 1 < 0. Conversely if j 0, j ޚ, and j < 0, then as ޚ C 2 C 2 we have j 1, so 0 j 1 1: Of course j < 0 also C D C means that j j j . Thus we can write equation (9.41) as D D j C j j S.n ; j k/ 4 .n / j cn 2 cj j C j C C I I1 C j C j D n k k j 0 r C p j X<0 (9.42) C for n ( n ) 1: C D THEOREM 9.43 (A REFORMULATION OF THEOREM 9.27). For a modular form f.z/ of weight zero with Fourier expansion given by equation (9.33), its n-th Fourier coefﬁcient cn is given by equation (9.42), for n 1. Here is C assumed to be a negative integer. Instead of applying Theorem 9.20 and taking r 0 there, without justiﬁcation, D physicists can now use Theorem 9.43 for a CFT modular invariant partition function, such as that of equation (9.4), and therefore stand on steady mathematical ground.

Appendix A. Uniform convergence of improper integrals. For the reader’s convenience we review the conditions under which an improper integral f.s/ 1 F.t; s/ dt D a defines a holomorphic function f.s/. In particular, a verification of the entirety R of the function J.s/ in equation (1.4) is provided. The function F.t; s/ is defined on a product Œa; / D with D ރ some 1 open subset, where it is assumed that 1 F.t; s/ dt exists for each s D — say a 2 t F.t; s/ is integrable on Œa; b for every b > a. Thus f.s/ is well-defined ‘ R on D. By definition, the integral f.s/ is uniformly convergent on some subset D0 D if to each " > 0 there corresponds a number B."/ > a such that for b > b B."/ one has F.t; s/ dt f.s/ <" for all s D0. An equivalent definition a 2 ˇR ˇ ˇ ˇ LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 79 is given by the following Cauchy criterion: f.s/ is uniformly convergent on D0 if and only if to each " > 0 there corresponds a number B."/ > a such that b2 F.t; s/ dt <" for all b2 > b1 > B."/ and all s D0. For clearly if f.s/ is b1 2 uniformly convergent on D0 and " > 0 is given, we can choose B."/ > a such ˇR b ˇ ˇthat F.t; s/ˇdt f.s/ < "=2 for b > B."/, s D0. Then for b2 > b1 > B."/ a 2 and s D0, we have ˇR2 ˇ ˇ ˇ b2 b2 b1 F.t; s/ dt F.t; s/ dt f.s/ F.t; s/ dt f.s/ ; D Zb1 Za Za which implies b2 F.t; s/ dt <"=2 "=2 ". Conversely, assume the alternate b1 C D condition. Define the sequence fn.s/ n>a of functions on D0 by ˇR ˇ f g ˇ ˇ n def fn.s/ F.t; s/ dt: D Za b2 Given ">0 we can choose, by hypothesis, B."/>a such that b1 F.t; s/ dt <" for b2 > b1 > B."/ and s D0. Let N."/ be an integer > B."/. Then for 2 ˇR ˇ integers n > m N."/ and for all s D0 we see that ˇfn.s/ fm.s/ˇ n 2 D F.t; s/ dt <". Therefore, by the standard Cauchy criterion, the sequence m ˇ ˇ fn.s/ n>a converges uniformly on D0 to a function g.s/ onˇD0: For any "1 >ˇ 0, ˇfR g ˇ ˇthere exists anˇ integer N."1/ > a such that for an integer n N."1/, one has n "1 > fn.s/ g.s/ F.t; s/ dt f.s/ for all s D0, since necessarily D a 2 g.s/ f.s/. Now let " > 0 be given. Again, by hypothesis, we can choose Dˇ ˇ ˇR ˇ b2 B."/ˇ > a such thatˇ forˇ b2 > b1 > B."/ oneˇ has that b1 F.t; s/ dt < "=2 for all s D0. Taking the quantity "1 considered a few lines above equal to 2 ˇ R ˇ "=2, we can find an integer N."1/ > B."1/ such that forˇ an integer n ˇN."1/, n "=2 "1 > F.t; s/ dt f.s/ for all s D0. Thus suppose b > N."1/. D a 2 Then, for all s D0, ˇR2 ˇ ˇ ˇ b N."1/ b F.t; s/ dt f.s/ F.t; s/ dt f.s/ F.t; s/ dt D C ˇ Za ˇ ˇ Za ZN."1/ ˇ ˇ ˇ ˇ " ˇ" ˇ ˇ ˇ ˇ "; ˇ ˇ ˇ 2 C ˇ2 D where N."1/ > a since B."1/ > a, with N."1/ dependent only on ". This shows that f.s/ is uniformly convergent on D0. The Cauchy criterion is therefore validated. As an example, we use the Cauchy criterion to prove the following, very useful result:

THEOREM A.1 (WEIERSTRASS M-TEST). Let M.t/ 0 be a function on def Œa; / that is integrable on each Œa; b with b > a. Assume also that I 1 D 80 FLOYDL.WILLIAMS

1 M.t/ dt exists. If F.t; s/ M.t/ on Œa; / D0, thenf.s/ 1F.t; s/ dt a j j 1 D a converges uniformly on D . Again D is any subset of D. R 0 0 R b PROOF. Let " > 0 be assigned. That I limb a M.t/ dt implies there Db !1 exists a number B."/ > a such that I M.t/ dt < "=2 for b > B."/. If a R b > b > B."/, then 2 1 ˇ R ˇ ˇ ˇ b2 b2 b1 " " M.t/ dt M.t/ dt I I M.t/ dt < " D C 2 C 2 D I ˇZb1 ˇ ˇZa Za ˇ ˇ ˇ ˇ ˇ ˇ b2 ˇ ˇ b2 b2 ˇb2 henceˇ F.t; s/ˇdt ˇ F.t; s/ dt M.t/ dt ˇ M.t/ dt (since b1 b1 b1 D b1 M.t/ 0; b2 > b1). But this is less than ", for all s D0. Theorem A.1 follows, ˇR ˇ R ˇ ˇ R 2 ˇR ˇ therefore,ˇ by the Cauchyˇ criterion.ˇ ˇ ˇ ˇ ˜ The question of the holomorphicity of f.s/ is settled by the following theorem.

THEOREM A.2. Again let F.t; s/ be defined on Œa; / D with D ރ an open 1 subset. Assume (i) F.t; s/ is continuous on Œa; / D (in particular for each s D, t F.t; s/ 1 2 ‘ is integrable on Œa; b for every b > a); (ii) for every t a fixed, s F.t; s/ is holomorphic on D; ‘ (iii) for every s D fixed, t @F.t; s/=@s is continuous on Œa; /; def 2 ‘ 1 (iv) f.s/ 1 F.t; s/ dt converges for every s D; and D a 2 (v) f.s/ converges uniformly on compact subsets of D. R Then f.s/ is holomorphic on D, and f .s/ 1 @F.t; s/=@sdt for every s D. 0 D a 2 Implied here is the existence of the improper integral @F.t; s/=@sdt on D. R a1 The idea of the proof is to reduce matters to a situationR where the integration n a1 over an infinite range is replaced by that over a finite range a , where holomorphicity is known to follow. This is easily done by considering again the R def n R sequence fn.s/ n>a discussed earlier: fn.s/ F.t; s/ dt on D, which is f g D a well-defined by (i). If K D is compact, then given (v), the above argument R with D0 now taken to be K shows exactly (by way of the Cauchy criterion) that fn.s/ n>a converges uniformly on K (to f.s/ by (iv)). On the other hand, by f g (i), (ii), (iii) we have that (i) F.t; s/ is continuous on Œa; n D; (ii) for every 0 0 t Œa; n fixed, s F.t; s/ is holomorphic on D, and (iii) for every s D 2 ‘ 0 2 fixed, t @F.t; s/=@s is continuous on Œa; n here a < n ޚ. Given (i)0, (ii)0 ‘ I 2 n and (iii)0, it is standard in complex variables texts that fn.s/ a F.t; s/ dt is n D holomorphic on D and that f .s/ @[email protected]; s/ dt. Since we have noted that n0 D a R the sequence fn.s/ n>a converges uniformly to f.s/ on compact subsets K of f g R D, it follows by the Weierstrass theorem that f.s/ is holomorphic on D, and that f .s/ f .s/ pointwise on D — with uniform convergence on compact subsets n0 ‘ 0 LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 81

n of D, in fact. That is, f 0.s/ limn fn0.s/ limn a @F.t; s/=@sdt D !1 D !1 D @F.t; s/=@sdt on D, which proves Theorem A.2. a1 R R As an application, we check that the function J.s/ in deﬁnition (1.4) is an def n2t entire function. First we claim that the function 0.t/ 1 e , for t > 0, D n 1 converges uniformly on Œ1; /. This is clear, by the WeierstrassD M-test, since 2 1 2 P n2t n for n; t 1, we have n n, hence n t nt n, hence e e , n and moreover n1 1 e is a convergent geometric series. Therefore 0.t/ is D n2t continuous on Œ1; /, since the terms e are continuous in t on Œ1; /. By P 1 def s 1 deﬁnitions (1.2), (1.4), J.s/ 1 F.t; s/ dt for F.t; s/ 0.t/t on Œ1; / ރ, D 1 D 1 where F.t; s/ therefore is also continuous. Again for n; t 1, n2t nt and 2 R also t , so e n t e nt and e t e , so 1 e t 1 e , so 1 1 e def C: 1 e t 1 e D e 1 D That is,

t 1 n2t 1 nt 1 t n e t 0.t/ e e .e / Ce D D D 1 e t n 1 n 1 n 1 XD XD XD t Re s bt a for t 1, so F.t; s/ Ce t on Œ1; / ރ, where 1 e t dt con- 1 1 verges for b > 0; a ޒ. Thus J.s/ converges absolutely for every s ރ. ˇ 2ˇ R 2 We see that conditionsˇ ˇ (i) and (iv) of Theorem A.2 hold. Conditions (ii) and (iii) certainly hold. To check condition (v), let K ރ be any compact subset. The continuous function s Re s on K has an upper bound Re s ! W on K t Re s t on Œ1; / K (since log t 0 for t 1). That is, on ) 1 t t Œ1; / K the estimate F.t; s/ Ce t holds where 1 e t dt < , 1 j j 1 1 implying that J.s/ converges uniformly on K, by Theorem A.1. Therefore J.s/ R is holomorphic on ރ by Theorem A.2. def B. A Fourier expansion (or q-expansion). The function q.z/ e2iz is holo- D morphic and it satisﬁes the periodicity condition q.z 1/ q.z/. Suppose f.z/ C D is an arbitrary holomorphic function deﬁned on an open horizontal strip

def S z ރ b1 < Im z < b2 (B.1) b1;b2 Df 2 j g as indicated in the ﬁgure at the top of the next page, where b1; b2 ޒ, b1 < b2. 2 Suppose also that f.z/ satisﬁes the periodicity condition f.z 1/ f.z/ C D on S ; clearly z S implies z r S for all r ޒ. Then f.z/ b1;b2 2 b1;b2 C 2 b1;b2 2 has a Fourier expansion (also called a q-expansion)

n 2inz f.z/ anq.z/ ane (B.2) D D n ޚ n ޚ X2 X2 82 FLOYDL.WILLIAMS

y b2 D

Sb1;b2

y b1 D

on S , for suitable coefficients an ރ; see Theorem B.7 and equation (B.6) b1;b2 2 below for an expression of the an. The finiteness of b2 is not essential for the validity of equation (B.2). In fact, one of its most useful applications is in case when S is the upper half plane: b1 0, b2 . The Fourier expansion b1;b2 D D 1 of f.z/ follows from the local invertibility of the function q.z/ and the Laurent expansion of the function .f q 1/.z/. We fill in the details of the proof. ı Note first that q.z/ is a surjective map of the strip Sb1;b2 onto the annulus def def def ރ 2b2 2b1 Ar1;r2 w r1 < w < r2 for r1 e > 0, r2 e > 0 (see Df 2 j j j g D D figure below). y

Ar1;r2

r1 x

r2 LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 83

2y For if z x iy Sb1;b2 , we have q.z/ e and b1 < y < b2, 2b1 D 2Cy 2 2b2 j j D so e > e > e , and w q.z/ Ar1;r2 . On the other hand, if D it 2 w Ar1;r2 is given choose t ޒ such that e w= w (since w 0), and deﬁne 2 2 defD j j ¤ r log w . Then one quickly checks that z t=2 ir=. 2/ S such D j j D C 2 b1;b2 that q.z/ w, as desired. D From f.z 1/ f.z/, it follows by induction that f.z n/ f.z/ for C D C D every positive integer n, and therefore for every negative integer n, f.z/ D f.z n . n// f.z n/; i.e. f.z n/ f.z/ for every n ޚ. Also since C C D 2izC1 2iz2 C 2Di.z1 z2/ 2 q.z1/ q.z2/ e e e 1 z1 z2 n for D () D () D () D C some n ޚ, the surjectivitiy of q.z/ implies that the equation F.q.z// f.z/ 2 D provides for a well-deﬁned function F.w/ on the annulus Ar1;r2 . To check that

F.w/ is holomorphic, given that f.z/ is holomorphic, take any w0 Ar1;r2 2 and choose z0 Sb1;b2 such that q.z0/ w0, again by the surjectivity of q.z/. 2 2iz D Since q .z/ 2ie implies in particular that q .z0/ 0, one can conclude 0 D 0 ¤ that q.z/ is locally invertible at z0: there exist " > 0 and a neighborhood N of z0, N S , on which q is injective with b1;b2 i:e: ރ q.N / N".q.z0// N".w0/ w w w0 <" Ar1;r2 ; D D Df 2 j j j g 1 and with q holomorphic on N".w0/. Thus, on N".w0/, F.w/ F q.q 1.w// .f q 1/.w/; D D ı which shows that F is holomorphic on N".w0/ and thus is holomorphic on

Ar1;r2 , as w0 Ar1;r2 is arbitrary. 2 Now F.w/ has a Laurent expansion

1 n 1 bm F.w/ anw Q D Q C wm n 0 m 1 XD XD on the annulus Ar1;r2 where the coefficients an; bm are given by Q Q 1 F.w/ dw 1 F.w/ an ; bm dw; Q D 2i wn 1 Q D 2i w m 1 Z C Z C for any circle in Ar1;r2 that separates the circles w r1, w r2. We j j Ddef j j D choose to be the circle centered at w 0 with radius R e 2b, given any D D b with b1 < b < b2; r1 < R < r2. For a continuous function .w/ on the change of variables v.t/ 2t on Œ0; 1 permits the expression D 2 1 .w/ dw .Reiv/Rieivdv 2i .Re2it / Re2it dt D 0 D 0 Z Z 1 Z 2i .e2i.t ib//e2i.t ib/ dt; D C C Z0 84 FLOYDL.WILLIAMS by definition of R. For the choices .w/ F.w/=wn 1, F.w/=w m 1, re- D C C spectively, one finds that

1 2i.t ib/ 2in.t ib/ an F.e /e dt; Q D C C 0 (B.3) Z 1 2i.t ib/ 2im.t ib/ bm F.e /e dt; Q D C C Z0 for n 0; m 1. However t ib S since b1 < b < b2 so by deﬁnition C 2 b1;b2 of F.w/ the equations in (B.3) are

1 2in.t ib/ an f.t ib/e dt; Q D C C 0 (B.4) Z 1 2im.t ib/ bm f.t ib/e dt; Q D C C Z0 for n 0; m 1, and moreover the Laurent expansion of F.w/ has a restatement

1 n 1 bm f.z/ anq.z/ Q (B.5) D Q C q.z/m n 0 m 1 XD XD on Sb1;b2 . One can codify the preceding formulas by deﬁning

1 def 2in.t ib/ an f.t ib/e dt (B.6) D C C Z0 for n ޚ, again for b1 < b < b2. Then an an for n 0 and a n bn for 2 D Q D Q n 1. By equation (B.5) we have therefore completed the proof of equation (B.2):

THEOREM B.7 (A FOURIER EXPANSION). Let f.z/ be holomorphic on the open strip Sb1;b2 deﬁned in equation (B.1), and assume that f.z/ satisﬁes the periodicity condition f.z 1/ f.z/ on S . Then f.z/ has a Fourier C D b1;b2 expansion on Sb1;b2 given by equation (B.2), where the an are given by equation (B.6) for n ޚ, for arbitrary b subject to b1 < b < b2. 2

Theorem B.7 is valid if Sb1;b2 is replaced by the upper half-plane C (with b1 0, b2 ), for example, as we have indicated. For clearly the preceding D D 1 arguments hold for b2 . Here, in place of the statement that q Sb1;b2 D 1 def 2b2 def 2b1 W ! Ar1;r2 is surjective (again for r1 e , r2 e ; b2 < ), one simply D D 1 employs the statement that q w ރ 0 < w < 1 is surjective. W C !f 2 j j j g LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 85

C. Poisson summation and Jacobi inversion. The Jacobi inversion formula (1.3) can be proved by a special application of the Poisson summation formula (PSF). The latter formula, in essence, is the statement

f.n/ f.n/; (C.1) D O n ޚ n ޚ X2 X2 for a suitable class of functions f.x/ and a suitable normalization of the Fourier transform f.O x/ of f.x/. The purpose here is to prove a slightly more general version of the PSF, which applied in a special case, coupled with a Fourier transform computation, indeed does provide for a proof of equation (1.3). For a function h.x/ on ޒ, the deﬁnition

def h.x/ 1 h.t/e 2ixt dt (C.2) O D Z1 will serve as our normalization of its Fourier transform. Here’s what we aim to establish:

THEOREM C.3 (POISSON SUMMATION). Let f.z/ be a holomorphic function def on an open horizontal strip Sı z ރ ı < Im z < ı ; ı > 0, say with 1 D f 2 j g f ޒ L .ޒ; dx/. Assume that the series 1 f.z n/, 1 f.z n/ j 2 n 0 C n 1 converge uniformly on compact subsets of S . ThenD for any z S D ıP 2Pı f.z n/ e2inzf.n/: (C.4) C D O n ޚ n ޚ X2 X2 In particular for z 0 we obtain equation (C.1). D PROOF. By the Weierstrass theorem, the uniform convergence of the series 1 f.z n/ and 1 f.z n/ on compact subsets of S means that the n 0 C n 1 ı functionD D P P def F.z/ f.z n/ 1 f.z n/ 1 f.z n/ D n ޚ C D n 0 C C n 1 P2 PD PD on Sı is holomorphic. F.z/ satisﬁes F.z 1/ n ޚ f.z n 1/ C D 2 C C D n ޚ f.z n/ F.z/ on Sı. Therefore, Theorem B.7 of Appendix B is 2 C D P applicable, where the choice b 0 is made (b1 ı; b2 ı): F.z/ has a P D D D Fourier expansion 2inz F.z/ ane (C.5) D n ޚ P2 def 1 2int ޚ on Sı, where an 0 F.t/e dt for n . Since Œ0; 1 Sı is compact, D 22int 2int for n ޚ ﬁxed the series 1 f.t l/e and 1 f.t l/e 2 R l 0 C l 1 (whose sum is F.t/e 2int ) convergeD uniformly on Œ0; 1 (byD hypothesis, given P P 86 FLOYDL.WILLIAMS

2int of course that e 1). Therefore an can be obtained by termwise inte- j jD gration; we start by writing

1 1 2int 1 2int an f.t l/e f.t l/e dt D 0 l 0 C C l 1 Z D D P1 P 1 1 2int 1 2int f.t l/e dt f.t l/e dt D l 0 0 C C l 1 0 D Z D Z P l 1 P l 1 1 C f.v/e 2in.v l/dv 1 C f.v/e 2in.v l/dv; D C C l 0 Zl l 1 Z l PD PD by the change of variables v.t/ t l and v.t/ t l on Œl; l 1, Œ l; l 1, ޚ D C D l 1 C 2int C respectively (with l ), This is further equal to l1 0 l C f.t/e dt l 1 22int 2int D 0 2int C 1 C f.t/e dt 1 f.t/e dt f.t/e dt l 1 l D 0 P C R D Df.t/e 2int dt f.n/, by deﬁnition (C.2). That is,1 by (C.5), for z S P1 R O R R ı 1 2inz D def 2 ޚ f.n/e F.z/ ޚ f.z n/, which concludes the proof of R n O D D n C Theorem2 C.3. 2 ˜ P P Other proofs of the PSF exist. In contrast to the complex-analytic one just presented, a real-analytic proof (due to Bochner) is given in Chapter 14 of [38], for example, based on Fejer’s´ Theorem, which states that the Fourier series of a continuous, 2-periodic function .x/ on ޒ is Cesaro` summable to .x/. def z2t As an example, choose f.z/ ft .z/ e for t > 0 ﬁxed. In this case D D f.z/ is an entire function whose restriction to ޒ is Lebesgue integrable; the restriction is in fact a Schwartz function. We claim that the series

1 f.z n/ and 1 f.z n/ n 0 C n 1 PD PD converge uniformly on compact subsets K of the plane. Since K is compact z2t the continuous functions z e and z Re z on ރ are bounded on K z2t ‘ ‘ W e M1, Re z M2 on K for some positive numbers M1; M2. Let n0 2 be an integer > 1 2M2. Then for n ޚ with n n0 one has n n.1 2M2/, ˇ ˇ2 ˇC ˇ 2 z2t .n2 2nz/t C ˇhence nˇ 2nMˇ2 nˇ, so that f.z n/ e e for z K. But C D C 2 2 2 2 e .n 2nz/t e .n 2n Re z/t e .n 2nM2/t e nt C D C D ˇz2t ˇ nt nt and e ˇ M1, soˇ f.z n/ M1e on K, with n1 0 Me j C j D clearly convergent for t > 0. Therefore, by the M -test, 1 f.z n/ con- ˇ ˇ n 0P C vergesˇ absolutelyˇ and uniformly on K. Similarly, for n ޚ, weD have f.z n/ z2t .n2 2nz/t 2 P2 D e e , where for n n0 and z K again n 2nM2 n, but 2 2 where we now use the bound Re z M2: n 2n Re z n 2nM2 n LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 87

tn f.z n/ M1e on K (for n n0), so n1 1 f.z n/ converges abso- j j D def z2t lutely and uniformly on K, which shows that ft .z/ e ; t > 0, satisﬁes P D the hypotheses of Theorem C.3. The conclusion

n2t e ft .n/ (C.6) D O n ޚ n ޚ X2 X2 is therefore safe, and the left-hand side here is .t/ by deﬁnition (1.2). One is therefore placed in the pleasant position of computing the Fourier transform

.C:2/ 1 y2t 2ixy ft .x/ e e dy; (C.7) O D Z1 which is a classical computation that we turn to now (for the sake of completeness). v2 2icv=pt For real numbers a; b; c; t with a < b, t > 0, note that e e 2 2 2 D e v e2icvp=t e c =t e .v icp=t/ . By the change of variables v.x/ D D ptx on Œapt; bpt, therefore,

b c2=t bpt x2t 2icx e .v icp=t/2 e e dx e dv: (C.8) D pt Za Zapt Next we show that for b ޒ 2 2 2 1 e .x ib/ dx 1 e x dx: (C.9) C D Z1 Z1 To do this, assume ﬁrst that b > 0 and consider the counterclockwise oriented rectangle CR of height b and width 2R CR C1 C2 C3 C4. W D C C C y C2

C4 88 FLOYDL.WILLIAMS

def z2 By Cauchy’s theorem, 0 IR e dz. Now D D CR z2 R x2 R C1 e dz R e dx; 2 D 2 2 2 e z dz i b e .R ix/ dx ie R b e 2xRiex dx; RC2 R 0 C 0 2 D 2 D : e z dz R e .x ib/ dx; RC3 R R C R 2 D 2 e z dz e z dz RC4 D RC2 R z2 RR2 b x2 Thus C2 e dz e 0 e dx, which tends to 0 as R . That is, x2 .x ib/2 ! 1 0 limR IR 1 e dx 1 e C dx, which proves equation D ˇR !1 Dˇ 1 R 1 .x ib/2 . x ib/2 (C.9)ˇ for b > 0. Ifˇ b < 0, write 1 e C dx 1 e C dx .x i. b//2 R x2 1R D 1 D 1 e dx 1 e dx by the previous case, since b > 0. Thus C D R R (C.9)1 holds for all b ޒ1(since it clearly holds for b 0). By (C.8) it then R 2 R D follows that R 1 x2t 2icx x2t 2icx e e dx lim e e dx D R R Z1 !1 Z c2=t Rpt e .x i. c/p=t/2 lim e C dx D pt R Rpt !1 Z c2=t c2=t e 1 x2 e e dx p: (C.10) D pt D pt Z1

PROPOSITION C.11. For c ޒ and t > 0, we have 2 c2=t 1 x2t 2icx 1 x2t 2icx e e e dx e e dx : D D pt Z1 Z1 x2=t Hence equation (C.7) is the statement that ft .x/ e =pt: O D Having noted that the left-hand side of equation (C.6) is .t/, we see that (C.6) n2=t def 1 (by Proposition C.11) now reads .t/ ޚ e =pt . /=pt, which D n D t proves the Jacobi inversion formula (1.3). 2 P D. A divisor lemma and a scholium. The following discussion is taken, nearly word for word, from [38] and thus it has wider applications — for example, applications to the theory of Eisenstein series (see pages 274–276 of that reference). For integers d; n with d 0 write d n, as usual, if d divides n, and ¤ j def write d - n if d does not divide n. For n 1; ރ let .n/ d 2 D 0

def Let an be a sequence of complex numbers such that a 1 an < . f gn1 1 D n 1 j j 1 We shall proveD a lemma to the effect that D P 1 1 1 m amn .n/an: (D.1) D n 1 m 1 n 1 XD XD XD Before formulating a precise statement of (D.1) it is useful to consider some simple observations. For m; k 1 in ޚ set m km .k/ def .k/ def sm akj ; tm d.k; l/al : D j 1 D l 1 PD PD .k/ .k/ We ﬁrst prove by induction that the m-th partial sum sm equals tm for every m. For m 1, s.k/ a . On the other hand, t .k/ a because an integer l in D 1 D k 1 D k the range 1 l k is a multiple of k if and only if l k. Proceeding inductively, .k/ .k/ .k/ D .k/ one has sm 1 sm ak.m 1/ tm ak.m 1/. On the other hand, tm 1 .k/ C D C C D C C C D tm d.k; km 1/akm 1 d.k; km 2/akm 2 d.k; km k/akm k. C C C C C C CC C C For l ޚ and 1 l k, we have k km l k l k l (again), so .k/ 2 .k/ .k/ j C () j () D tm 1 tm akm k sm 1, which completes the induction. C D C C D C .k/ .k/ Now take m in the equality sm tm to conclude that 1 a ! 1 D j 1 kj exists and D P 1 1 a d.k; l/a : (D.2) kj D l j 1 l 1 XD XD Re If Re < 1, then since d.k; n/k an an =k , we have j j j j 1 d.k; n/k an an . Re / k 1 j j j j D Ps (where .s/ n1 1 1=n , Re s >1, is the Riemann zeta function) and moreover D D the iterated series n1 1 k1 1 d.k; n/k an converges: P D D j j

P1 P1 d.k; n/k an a. Re /: (D.3) j j n 1 k 1 XD XD By elementary facts regarding double series (found in advanced calculus texts) it follows that one can conclude that the double series n1 1 k1 1 d.k; n/k an converges absolutely, and that D D P P 1 1 1 1 d.k; n/k an d.k; n/k an : (D.4) D n 1 k 1 k 1 n 1 XD XD XD XD Similarly for k 1 ﬁxed, equation (D.2) (with an n1 1 replaced by an n1 1) Re f Reg D fj jg D yields m1 1 k akm k m1 1 akm k l1 1 d.k; l/ al . That is, D j jD D j jD D j j P P P 90 FLOYDL.WILLIAMS

m1 1 k akm < and moreover the iterated series k1 1 m1 1 k akm D j j 1 Re D D j j converges, as it equals k1 1 k l1 1 d.k; l/ al a. Re /. Thus, sim- P D D j j P P ilarly to equation (D.4), one has that the double series k1 1 m1 1 k akm converges absolutely, andP equality ofP the corresponding iteratedD seriesD prevails: P P 1 1 1 1 k akm k akm : (D.5) k 1 m 1 D m 1 k 1 PD PD PD PD Given these observations, we can now state and prove the main lemma regarding the validity of equation (D.1):

DIVISOR LEMMA. Let an n1 1 be a sequence of complex numbers such that the f g D series n1 1 an converges. Let .n/ 0 n. ThatD is, j P P 1 d.k; n/k .n/ (D.6) D k 1 XD ރ for any . The series n1 1 Re .n/ an is, by (D.6), the iterated series 2 Re D j j n1 1 k1 1 d.k; n/k an , which we have seen converges and is bounded D D jPj above by . Re / according to (D.3). Clearly .n/ Re .n/, so that P P j j also n1 1 .n/ an converges. Again by (D.6), we have n1 1 .n/an D j jj j D D n1 1 k1 1 d.k; n/k an . Now apply equations (D.4), (D.2), (D.5), suc- DP D P cessively, to express the latter iterated series as k1 1 n1 1 d.k; n/k an P P D D D k1 1 k m1 1 akm m1 1 k1 1 k akm , which proves (D.1). We D D D D D P P have already seen that the double series m1 1 n1 1 m amn which equals P P P P D D k1 1 m1 1 k akm converges absolutely. By equation (D.5), then one de- D D P P rives the equality of the corresponding iterated series m1 1 n1 1 m amn P P D D ˜ and n1 1 m1 1 m amn . D D P P .k/ .k/ GoingP backP to the equalitysm tm of the previous page, we actually have D the following fact, recorded for future application: LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 91

ޚ SCHOLIUM. Given a sequence of complex numbers an n1 1 and k with f g D 2 k 1, the series 1 a converges if and only if the series 1 d.k; j /aj j 1 kj j 1 converges, in which caseD these series coincide. D P P As an example, we use the Divisor Lemma to prove the next lemma, which is important for Lecture 4.

LEMMA D.7. Fix z; k ރ with Im z > 0, Re k > 2. Then the iterated series k 1 22imn k 1 2inz m1 1 n1 1 n e exists, the series n1 1 1 k.n/n e con- vergesD absolutelyD , and D P P P 1 1 k 1 2imn 1 k 1 2inz 1 2inz n e 1 k.n/n e k 1.n/e : (D.8) m 1 n 1 Dn 1 Dn 1 D D D D P P P P PROOF. The last equality comes from .n/ n .n/. To show the ﬁrst, def k 1 2inz D Rek 1 2n Im z set an n e . Then n1 1 an n1 1 n e converges D D j jD 1 k D 1 k k 1 2imnz by the ratio test, since Im z > 0. Also m amn m .mn/ e P P D nk 1e2imnz, where Re k > 2 Re.1 k/ < 1. By the Divisor Lemma D ) (for 1 k), the series n1 1 1 k.n/an converges absolutely, the iterated D 1 k D 1 k series n1 1 m1 1 m amn converges, and n1 1 m1 1 m amn D D P D D D˜ n1 1 1 k.n/an. Substituting the value of an proves the desired equality. D P P P P PAs another example, consider the sequence an n1 1 given by f g D def 1 2nxi s 2 an e˙ K 1 .2ny/n s 2 D for x; y ޒ, y > 0, s ރ ﬁxed, where 2 2 def 1 1 z 1 1 K.z/ exp t t dt (D.9) D 2 0 2 C t Z is the K-Bessel function for Re z > 0, ރ. To see that 1 an < , one 2 n 1 j j 1 applies the asymptotic result D P t lim ptK.t/e : (D.10) t D 2 !1 r t In particular, choose M >0 such that ptK.t/e =2 < =2 for t >M; t that is, K.t/ < 2 =2te for t > M. Then if Ns;y is an integer with j j ˇ p ˇ p Ns;y > M 1 =2y we see that for nˇ Ns;y, 2ny > M ˇ 1 , so s 2 s 2 p 1=2 2ny n 2ny K 1 .2ny/ < 2 e e s 2 2 2ny D py I r ˇ ˇ Re s 1 2ny Re s 1 2ny therefore aˇn <.n =ˇpy/ e , where 1 n e converges j j n 1 by the ratio test since y > 0. D P 92 FLOYDL.WILLIAMS

def Now assume that Re s > 1 so that 1 2s 1 satisfies Re < 1. Also def 2mnxi n Ds 2 C m amn e˙ K 1 .2mny/ for m; n 1. The Divisor Lemma D s 2 m gives 1 s 2 1 1 2mnxi n e˙ K 1 .2mny/ s 2 m 1 n 1 m D D 1 P P 1 2nxi s 2 2s 1.n/e˙ K 1 .2ny/n ; (D.11) s 2 D n 1 C PD with absolute convergence of the latter series, and convergence of the iterated series which coincides with the iterated series 1 s 2 1 1 2mnxi n e˙ K 1 .2mny/ : s 2 n 1 m 1 m D D E. Another summationP P formula and a proof of formula (2.4). In addition to the useful Poisson summation formula f.n/ f.n/ (E.1) D O n ޚ n ޚ X2 X2 of Theorem C.3, there are other very useful, well known summation formulas. The one that we consider here assumes the form f.n/ the sum of residues of . cot z/f.z/ at the poles of f.z/; (E.2) n ޚ D P2 for a suitable class of functions f.z/. As we applied formula (E.1) to a specific 2 function (namely the function f.z/ e z t for t > 0 fixed) to prove the Jacobi D inversion formula (1.3), we will, similarly, apply formula (E.2) to a specific function (namely the function f.z/ .z2 a2/ 1 for a > 0 fixed) to prove D C formula (2.4) of Lecture 2. The main observation towards the proof of formula (E.2) is that there is a nice bound for cot z on a square CN with side contours j j RN ; LN and top and bottom contours TN ; BN , as illustrated on the next page, for a fixed integer N > 0. The bound, in fact, is independent of N . Namely, cot z max.1; B/ < 2 (E.3) j j def on CN , for B .1 e /=.1 e /. We begin by checking this known result. D C For z x iy, x; y ޒ, we have iz y ix, and simple manipu- D C 2 D C lations give eiz e iz e yeix eye ix cot z i C i C : (E.4) D eiz e iz D e yeix eye ix Hence e y ey e y ey cot z C C j j e yeix eye ix e y ey j j j j LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 93

y TN .N 1 /; N 1 N 1 ; N 1 C 2 C 2 C 2 C 2

N 1 C 2 x N N 1 C

.N 1 /; .N 1 / N 1 ; .N 1 / C 2 C 2 C 2 C 2 BN

(since a b a b for a; b ރ). Since a a for a ޒ, this becomes j j j jj j 2 j j˙ 2 ˇ ˇ e y ey ˇ ˇ cot z C : (E.5) j j .e y ey/ ˙ Suppose (in general) that y > 1 . Then 2y >, so e 2y 2 1 e , which, by the choice of the minus sign in (E.5), allows us to write y y y 2y e e e e 1 e 1 def cot z C C < C B; j j ey e y e y D 1 e 2y 1 e D for y > 1 . Similarly if y < 1 , then 2y < , so e2y < e and 1 e2y > 2 2 1 e , and by the choice of the plus sign in (E.5) we get e y ey ey 1 e2y 1 e cot z C C < C B: j j e y ey ey D 1 e2y 1 e D Thus we see that def 1 e cot z < B C (E.6) j j D 1 e ރ 1 1 for z with either Im z > 2 or Im z < 2 . In particular on TN , Im z 12 1 1 1 D N > and on BN , Im z .N / < so by (E.6) the estimate C 2 2 D C 2 2 cot z < B (E.7) j j holds on both the contours TN and BN . 94 FLOYDL.WILLIAMS

1 Regarding the contours RN and LN we have z N 2 iy on RN and 1 2izD C 2Cy ޚ z .N 2 / iy on LN . On RN we have e e (since N ) D C C2iz 2y D 2 and, similarly, e e on LN . D Now consider the ﬁrst equation in (E.4) and multiply the fraction by 1 iz iz D e =e . On both RN and LN this leads to e2iz 1 e 2y 1 cot z i C i C ; D e2iz 1 D e 2y 1 and we conclude that 1 e 2 Im z 1 e 2 Im z cot z j j C 1 (E.8) j jD 1 e 2 Im z 1 e 2 Im z D C C on both contours RN and LN . The inequalities (E.7), (E.8) therefore imply (E.3), as desired, where we note that et 1 t for t ޒ, so e 1 > 3 C 2 C ) 2.e 1/ .e 1/ e 3 > 0. That is, indeed C D e 1 e 1 e def 2 > C C B: e 1 e D 1 e D We note also that sin z 0 z n ޚ. That is, since (again) N ޚ we D () D 2 2 cannot have sin z 0 on CN ; in particular CN avoids the poles of cot z D D cos z=sin z, and cot z is continuous on CN . Consider now a function f.z/ subject to the following two conditions:

C1. f.z/ is meromorphic on ރ, with only ﬁnitely many poles z1; z2;:::; zk, none of which is an integer. C2. There are numbers M; > 0 such that f.z/ M= z 2 holds for z >. j j j j j j Then: N THEOREM E.9. limN n N f.n/ exists and equals minus the sum of the !1 D residues of the function f.z/ cot z at the poles z1; z2;:::; zk of f.z/. One can replace condition C2, inP fact, by the more general condition (E.11) below.

PROOF. Since the poles zj are ﬁnite in number we can choose N sufﬁciently large that CN encloses all of them. The function cot z has simple poles at the integers (again as sin z 0 z n ޚ) and the residue at z n ޚ is D () D 2 def D 2 immediately calculated to be 1. Therefore the residue of .z/ f.z/ cot z D at z n ޚ is f.n/. As none of the zj are integers (by C1) the poles of .z/ D 2 within CN are given precisely by the set zj ; n 1 j k; N n N; n ޚ . f j 2 g By the residue theorem, accordingly, we deduce that 1 .z/dz 2i D ZCN N the sum of the residues of .z/ at the zj f.n/: (E.10) C n N DP LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 95

Now if lim .z/dz 0; (E.11) N CN D !1 Z we can let N in equation (E.10) and conclude the validity of Theorem E.9 ! 1 (more generally, without condition C2). 1 We check that condition (E.11) is implied by condition C2. Since z N 2 1 j j C on CN , we have for N > and z on CN the bound C 2 M M f.z/ : 2 2 j j z N 1 j j C 2 By the main inequality (E.3), we have cot z < 2 on CN , so .z/ 1 2 j j j j D f.z/ cot z < 2M= N on CN . Given that the length of CN is j j C 2 4 2 N 1 we therefore have the following estimate (for N 1 >): C 2 C 2 2M 32M .z/dz 8 N 1 ; (E.12) 1 2 2 CN C D 2N 1 ˇZ ˇ N 2 ˇ ˇ C C ˇ ˇ where we noteˇ that .z/ is continuousˇ on CN , because, as seen, cot z is continuous on CN . The inequality (E.12) clearly establishes the condition (E.11), by which the proof of Theorem E.9 is concluded. ˜ As an example of Theorem E.9 we choose 1 1 f.z/ D z2 a2 D .z ai/.z ai/ C C for a > 0 fixed. Hence f is meromorphic on ރ with exactly two simple poles def def z1 ai, z2 ai. Suppose, for example, that z > p2a: Then D D j j a2 1 a2 a2 z2 1 1 > ; so 1 1 ; so < 2: z 2 2 C z2 z 2 z2 a2 D a2 ˇ ˇ ˇ ˇ 1 j j ˇ ˇ j j ˇ C ˇ 2 ˇ ˇ ˇ ˇ C z ˇ ˇ ˇ ˇ ˇ ˇ Therefore f.z/ < 2= z 2; that is, f.z/ satisfies conditions C1, C2ˇ with Mˇ 2, j j j j def ˇ ˇ D p2a. The residue of .z/ f.z/ cot z at z1 is D D cot z cot ai lim .z z1/.z/ lim coth a; z z1 D z ai z ai D 2ai D 2a ! ! C since cos iw cosh w, sin iw i sinh w. Similarly, the residue of .z/ at z2 D D is .=2a/ coth a. As f. z/ f.z/, N f.n/ 1 2 N 1 . D n N D a2 C n 1 n2 a2 Theorem E.9 therefore gives D D C P P 1 1 1 2 coth a coth a coth a; a2 C n2 a2 D 2a 2a D a n 1 XD C 96 FLOYDL.WILLIAMS which proves the summation formula (2.4). def F. Generators of SL.2; ޚ/. Let SL.2; ޚ/. To prove that the elements def 1 1 def 0 1 D T and S generate , we start with a lemma. D 0 1 D 1 0 2 a b LEMMA F.1. Let c d with c 1. Then is a finite product 1 l , D 2 n m where each j has the form j T j S j for some nj ; mj ޚ. Here for a 2 D def 2 group element g and 0 > n ޚ, we have set gn .g 1/ n. 2 D The proof is by induction on c. If c 1, then 1 det ad bc ad b, D D D D so b ad 1, so D a ad 1 1 a 0 1 1 d 1 0 1 2 D 1 d D 0 1 1 0 0 1 0 1 D 1 a 0 1 a 1 1 d 1 0 d 0 for 1 T S , 2 T S . Proceeding by D 0 1 1 0 D D 0 1 0 1 D induction, we use the Euclidean algorithm to write d qc r for q; r ޚ with D C 2 0 r < c 2, say. If r 0, then 1 det ad bc .aq b/c, which D D D D shows that c is a positive divisor of 1. That is, the contradiction c 1 implies D that r > 0. Now

q a b 1 q a aq b a aq b T c d 0 1 c cq Cd c rC ; D D C D q a aq b 0 1 aq b a so T S c rC 1 0 rC c , which equals 1 l by induc- D D nj mj tion (since 1 r < c), where each j has the form j T S for some D nj ; mj ޚ. Consequently, 2

1 q nl ml 1 q 0 1 l S T . 1 l 1/.T S /T S ; D D which has the desired form for and which therefore completes the induction and the proof of Lemma F.1.

THEOREM F.2. The elements T; S generate : Every is a ﬁnite product nj mj 2 1 where each j has the form j T S for some nj ; mj ޚ. l 2 D 2 a b PROOF. Let be arbitrary. If c 0, then 1 det ad, so D c d 2 D D D a d 1, so D D˙ 1 b b 0 1 b 1 b 1 0 b 2 0 1 T S or 0 1 0 1 0 1 T S : D D D D D Since the case c 1 is already settled by Lemma F.1, there remains only the case 2 1 0 a b c 1. Then S 0 1 c d 1 l , by Lemma F.1 (since D D D 0 2 c 1), where the j have the desired form. Thus 1 .T S /, as D l desired. ˜ LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 97

G. Convergence of the sum of m ni ˛ for ˛ > 2. To complete the argument j C j that the Eisenstein series Gk.z/ converge absolutely and uniformly on each of the strips S in deﬁnition (4.5) (for k 4; 6; 8; 10; 12;::: ), we must show, A;ı D according to the inequality (4.10), that the series

def 1 S.˛/ ˛ D 2 m ni .m;n/ ޚ j C j X2 converges for ˛ > 2, where ޚ2 ޚ ޚ .0; 0/ . D f g For n 1; n ޚ, let n denote the set of integer points on the boundary of 2 the square with vertices .n; n/, . n; n/, . n; n/, .n; n/. As an example, 3 is illustrated here, with 24 8 3 points. D y

.i/ In general n has n 8n points. Also the n partition out all of the j j D nonzero integer pairs:

1 ޚ ޚ .0; 0/ n (G.1) f gD n 1 SD is a disjoint union.

1 8 LEMMA G.2. for ˛ 0; n 1. a bi ˛ n˛1 .a;b/ n j C j P2 PROOF. For .a; b/ n, either a n or b n, according to whether .a; b/ 2 D˙ D˙ lies on one of the vertical sides of the square (as illustrated above for 3), or on one of the horizontal sides, respectively. Thus a2 b2 either n2 b2 or C D C 98 FLOYDL.WILLIAMS

2 2 2 2 2 2 2 2 2 a n a b n . That is, for .a; b/ n we have a bi a b n , C ) C 2 j C j D C so a bi ˛ n˛ (since ˛ 0). Inverting and summing we get j C j 1 1 8n (by (i)) a bi ˛ n˛ n˛ .a;b/ n j C j .a;b/ n X2 X2 8 ; ˛ 1 D n which proves Lemma G.2. ˜ Now use (G.1) and Lemma G.2 to write

1 1 1 8 S.˛/ (G.3) D a bi ˛ n˛ 1 n 1 .a;b/ n n 1 XD X2 j C j XD for ˛ 0, which proves that S.˛/ < for ˛ 1 > 1, as desired. 1 References

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FLOYD L. WILLIAMS DEPARTMENT OF MATHEMATICS AND STATISTICS LEDERLE GRADUATE RESEARCH TOWER 710 NORTH PLEASANT STREET UNIVERSITY OF MASSACHUSETTS AMHERST, MA 01003-9305 UNITED STATES [email protected]