Universita` degli Studi di Milano Dipartimento di Matematica Dottorato di Ricerca in Matematica IX Ciclo Tesi di Dottorato

L-functions: Siegel-type theorems and structure theorems

Giuseppe Molteni

Relatore:

Alberto Perelli (Dipartimento di Metodi e Modelli Matematici, Universit`adi Genova) Giuseppe Molteni Dipartimento di Matematica Universit`adegli Studi di Milano Via Saldini, 50 20133 Milano — Italy email: [email protected] Acknowledgments

I wish to thank my thesis advisor, Prof. Alberto Perelli. During these years he has been an helpful teacher and a sure guide.

I wish to thank Prof. C. Viola and R. Dvornicich of University of Pisa, Prof. U. Zannier of University of Venezia and Prof. J. Kaczorowski of University of Pozna`nfor many interesting and useful discussions.

I wish to thank dr. A. Languasco of University of Padova and dr. A. Zaccagnini of University of Parma for their suggestions and friendship.

Contents

Notation iii Introduction 1 Chapter 1. Arithmetical relations coming from Euler products 3 1.1. Explicit computations 3 1.2. An Ω-result 5 1.3. An interesting identity 6 1.4. Estimates for products of coefficients of Dirichlet series with Euler product 9 1.5. The ”fudge factor” and Rankin-Selberg convolution 13 Chapter 2. The Siegel zero 21 2.1. Introduction 21 2.2. The axiomatic L∗-classes 26 2.3. Siegel-type theorems 44 Chapter 3. Existence of a singularity for certain functions of degree 1 55 3.1. Introduction 55 3.2. Definitions and results 55 3.3. Proof of lemma 58 3.4. Appendix 62

Chapter 4. About the Selberg class Sd, 0 ≤ d ≤ 1 65 4.1. Introduction and results 65 4.2. First proof 66 4.3. Second proof 67 Bibliography 69

i

Notation

s = σ + it with s ∈ C, σ =

f(x)  g(x) for x → x0 means that there exist positive constants c1 and c2 such that c1|f(x)| ≤ |g(x)| ≤ c2|f(x)| in a neighborhood of x0. (Z/qZ)∗ group of invertible elements of Z/qZ. (a, b) for a, b ∈ N, the greatest common divisor of a and b. χ Dirichlet character.

χ0 principal character: χ0(n) = 1 for every n ∈ N. τ(χ) Gauss sum for the character χ.

χD Kronecker symbol; when D is a fundamental discriminant, the Kronecker sym- bol is a real primitive character modulo D. d degree of a polynomial Euler product.

sn(α) the n-th elementary symmetric polynomial of the α1, . . . , αd variables: s (α) := P α ··· α . n 1≤j1<···

QA, WA main parameter and weight of L(s, A).

dA, dA arithmetic and analytic degree of L(s, A).

iii iv NOTATION

PA,B exceptional set for the Rankin-Selberg convolution L(s, A ⊗ B).

Λn(A) generalized von Mangoldt function. ¯ RA regular component of L(s, A ⊗ A). sym2 sym2 symmetric square map: GL(2) → GL(3), diag (α, β) −→ diag(α2, αβ, β2).

Cd a class of L-functions with arithmetic degree d.

S, Sd Selberg class and subset of elements with analytic degree d. ] ] S , Sd a class of L-functions similar to S and Sd respectively, without Euler product. R σ>a ds integral over a vertical line σ = σ0 with σ0 > a. a b Γ0(N) Hecke congruence group of level N: {( c d ) ∈ SL(2, Z), with c ≡ 0(N)}. Introduction

The theory of L-functions provides powerful methods to study various arithmetical problems, and hence several such functions have been associated with arithmetical, alge- braic and geometric objects. The Hecke and Artin L-functions associated with algebraic number fields, the L-functions associated with automorphic representations of GL(r) and the L-functions of elliptic curves are some examples. It has been observed that most L-functions arising in number theory share some basic analytic properties, namely: ∗ when normalized, a representation as Dirichlet series ∞ X −s L(s) := ann n=1 exists for 1, and a representation as Euler product Y L(s) = Lp(s) p holds, where p runs over the primes; ∗ L(s) has analytic continuation to C as a meromorphic function with a pole at s = 1 at most, and satisfies a functional equation of the following standard type Qsγ(s)L(s) = ωQ1−sγ¯(1 − s)L¯(1 − s), γ(s) being a product of Γ-factors, Q > 0 and |ω| = 1. The well known Selberg class S, see [56], and the L-class introduced by Carletti, Monti-Bragadin and Perelli [7] are attempts to axiomatically define a class including most L-functions of number-theoretical interest. In S and L the above properties of the L-functions are postulated, but some further conditions are imposed. For example, in S  the Ramanujan hypothesis in the form an  n is assumed, and in L ⊂ S, moreover, the Euler product has a polynomial form, i.e.,

−1 −s Lp(s) = Pp (p )

with Pp(x) ∈ C[x]. The Ramanujan hypothesis is conjectured for every L-function of number-theoretical interest, but it is not proved in general. Therefore, it is interesting to investigate an axiomatic class of L-functions where such hypothesis is replaced by weaker conditions. This will be done in Chapter 2, in the framework of certain L∗-classes. 1 2 INTRODUCTION

The results that in L are proved independently of the Ramanujan hypothesis extend almost immediately to L∗. However, the usual approach to some important results, such as upper-bounds of type L(1)  Q , depends heavily on the Ramanujan hypothesis: new techniques are necessary in the setting of L∗-classes. An essential ingredient of these techniques is the particular form of the multiplicative law satisfied by the coefficients an, depending on the polynomial form of the Euler product. In Chapter 1, a preliminary chapter for the whole thesis, this law is extensively investigated. A classical zero-free region holds for L∗-classes and, as in the original case of Dirichlet L-functions, the ”Siegel zero phenomenon” arises. Actually, it is conjectured that the Siegel zero β˜ does not exist, and this has recently been proved for cuspidal automor- phic forms on GL(2) and GL(3) by Goldfeld, Hoffstein and Lieman [20], Hoffstein and Ramakrishnan [25] and Banks [1]. Moreover, several results show that its existence is an extraordinary event. Therefore, it is an important and non-trivial problem to find a ”Siegel zero-free region”, as large as possible. For the classical Dirichlet L-functions, the best result in this direction is the famous Siegel theorem, stating that 1 − β˜  q− . In Chapter 2 we prove Siegel-type theorems for L∗-classes. These theorems ex- tend several known results, such as the classical Siegel theorem, that of Golubeva and Fomenko [21, 22] for holomorphic cusp forms and Hoffstein and Lockhart’s [24] result for the symmetric square of a Maass forms. Moreover, new results for the symmetric cube and other L-functions connected with holomorphic or Maass cusp forms are deduced from general principles. In Chapter 3, another class, denoted by C and inspired by the work of Duke and Iwaniec [14], is defined and investigated. In C, the exact form of the functional equation is not important and only some conditions about the growth on vertical lines are assumed. Moreover, the existence of the twisted functions ∞ X −s Lχ(s) := χ(n)ann χ primitive Dirichlet character n=1 is assumed. A polynomial Euler product is also assumed, so that Cd denotes the subclass of C of the elements with deg Pp(x) ≤ d, for every p. In C we prove an interesting phenomenon: the existence of an L-function L(s) of Cd, for some d ≥ 2, for which Lχ(s) is entire for every primitive character, is an obstruction to the existence of an analogous L-function of C1. In other words: every element of C1 has a twist with a pole at s = 1. This agrees with the conjecture that the Dirichlet L-functions and their shifts exhaust C1. Finally, it is well known that in the Selberg class an analytic notion of degree can be introduced; it is an important and unproved conjecture that the degree actually assumes only integer values. In accordance with this conjecture, it has been proved by several authors that the Selberg class Sd, the subset of functions with analytic degree d, is empty for 0 < d < 1. In Chapter 4 we give two new proofs of this result. CHAPTER 1

Arithmetical relations coming from Euler products

The Dirichlet series arising from number theory usually have an Euler product of polynomial type, i.e., they satisfy an identity of the form ∞ d −1 X an Y Y αj(p) L(s) = = 1 − ns ps n=1 p j=1

in some right half-plane σ > σ0. For example, this is true for the Dirichlet L-functions, the Hecke L-series with gr¨ossencharacter, the Artin L-functions, the automorphic (holo- morphic and Maass) L-functions and so on. This particular form of the Euler product allows proving, by algebraic tools, many relations on the coefficients an. Such relations are very useful when investigating cer- tain analytic properties of the L-functions. Therefore, in this chapter some arithmetical ∞ properties of the sequence {hk}k=0, defined by the identity d ∞ Y −1 X k (1.1) (1 − αjx) =: hkx , j=1 k=0

are investigated, and certain properties of the sequence an are deduced. Of course, the product on the left hand side of (1.1) arises from the p-th Euler factor of L(s). Actually, only Sections 1.2, 1.4 and Subsection 1.5.3 contain original results that are necessary for the main purposes of this thesis. In the other sections we present some results that appear to be original by themselves (Section 1.3) or by their proofs (Sections 1.1 and 1.5).

1.1. Explicit computations From definition (1.1) we have

X a1 ad (1.2) hk = α1 ··· αd , a1+···+ad=k ai≥0

but this expression of hk is not very satisfactory. Let X sn := αj1 ··· αjn for n = 1,. . ., d,

1≤j1<···

denote the elementary symmetric polynomials of the αj variables. Then the identity ∞ d d X k (1 − s1x + ··· + (−1) sdx ) hkx = 1 k=0 3 4 1. ARITHMETICAL RELATIONS COMING FROM EULER PRODUCTS

gives the recursive relations  h − s h + s h + ··· + (−1)ds h = 0 if k > 0,  k 1 k−1 2 k−2 d k−d (1.3) h0 = 1,  hk = 0 if k < 0 .

The recursion can be solved as follows: denoting by vn the column vector t vn = (hn, hn−1,. . .,hn−d+1) , (1.3) is equivalent to ( v = (1, 0, . . ., 0)t s −s s ... (−1)ds  0 with A := 1 2 3 d , vn = Avn−1 Id−1 0

where Id−1 is the identity matrix of order d − 1. It is well known that α1, ··· αd are the d−1 d−2 t eigenvalues of A with wj := (αj , αj , ··· , 1) as eigenvectors, so A is diagonalizable if we suppose αi 6= αj for every i 6= j. In this case we set M := (w1 ··· wd) so that G := −1 n −1 M AM is diagonal, G = diag(α1, ··· αd). Hence vn = MG M v0 and, if V (c1, ···, cd) Q denotes the Vandermonde determinant 1≤i

j d ∨ d k+d−1 X V (α1, ···, αd) X αj (1.4) h = αk+d−1(−1)j+1 = . k j V (α , ···, α ) Q (α − α ) j=1 1 d j=1 i6=j i j

In the general case suppose α1, ··· , αl distinct and let mi := #{j : αj = αi} for i = 1, ··· , l. Then (1.2) can be written as     X a1 al X X (1.5) hk = α1 ··· αl 1 ··· 1 .

a1+···+al=k c1+···+cm1 =a1 c1+···+cml =al ai≥0 ci≥0 ci≥0 But X a+m−1
1 = m−1 =: Pm(a)

c1+···+cm=a ci≥0 d
is a polynomial in a of degree m − 1. Hence defining the differential operators Pm α dα d by the substitution a −→ α dα into Pm(a), the equality  d  P α αa = P (a)αa m dα m holds by linearity and by the elementary identity  d k α αu = ukαu for every u, k ≥ 0. dα Therefore, (1.5) becomes     ∂ ∂ X a1 al (1.6) hk = Pm1 α1 ··· Pml αl α1 ··· αl . ∂α1 ∂αl a1+···+al=k ai≥0 1.2. AN Ω-RESULT 5

We substitute (1.4) in (1.6) obtaining

l k+l−1  ∂   ∂  X αj (1.7) h = P α ··· P α , k m1 1 ∂α ml l ∂α Q (α − α ) 1 l j=1 i6=j i j which finally gives the relation

l X k (1.8) hk = pj(k)αj , j=1 where pj(k) is a polynomial of degree ≤ mj − 1 in k. We prove that deg pj(k) = mj − 1. It is sufficient proving that the coefficient of m1−1 k k α1 in (1.7) is non-zero. But this coefficient is  ∂   ∂  1 αl−1P α ··· P α = 1 m2 2 ∂α ml l ∂α Ql 2 l i=2(αi − α1) l l l l−1 Y  ∂  1 Y  ∂  1 Y −1 = α1 Pmi αi = Pmi xi = , ∂α α − α ∂x x − 1 (1 − x )mi i=2 i i 1 i=2 i i i=2 i

where xi := αi/α1 6= 1 by our hypothesis, and hence this expression is obviously non-zero.

1.2. An Ω-result Using the results of previous section we can prove the following

Proposition 1.2.1. Assume that 0<|αj| ≤1 for every j, |αj| = 1 for some j and let M−1 mi := #{j : αj = αi, with |αi| = 1}, M := max {mi}. Then hk = Ω(k ); in particular, hk = Ω(1).

Proof. Let us consider (1.8). Clearly, the terms with |αj| < 1 are o(1), and we know that in (1.8) there are terms of order kM−1. Without loss of generality we may assume that these terms correspond to j = 1, ··· , l. Then

l X 1 h = kM−1 r eikθj + O , k j k j=1

for some real θj with θi 6= θj for i 6= j, rj 6= 0. Hence Proposition 1.2.1 follows if we prove Pl ikθj that Rk := j=1 rje 6→ 0 as k → ∞. By contradiction, let us assume that Rk → 0. −ikθ1 Then Rke → 0 as well, and hence by Ces`aro mean we have

N l N 1 X X 1 X  1  o(1) = R e−ikθ1 = r eik(θj −θ1) = r + O as N → ∞, N k j N 1 N k=1 j=1 k=1

a contradiction.  6 1. ARITHMETICAL RELATIONS COMING FROM EULER PRODUCTS

1.3. An interesting identity

We give here the deduction of a formula, identity (1.12) below, for the coefficients hk of the p-th Euler factor of L(s, symm f), where f is a normalized holomorphic newform for SL(2, Z) and symm is the morphism symm : GL(2) −→ GL(m + 1) , symm diag(α, β) −→ diag(αm, αm−1β, ··· , αβm−1, βm). This formula will not be used in the thesis; it is presented here since it appears to be of independent interest. We start observing that, introducing the polynomials N N N α1 α2 ··· αu u−2 u−2 u−2 α1 α2 ··· αu u−3 u−3 u−3 α1 α2 ··· αu (1.9) Du(N) = Du(α,N) := . . . , . . .

α α ··· α 1 2 u 1 1 ··· 1 identity (1.4) can be formulated as

Dd(k + d − 1) (1.10) hk = . Dd(d − 1) d m−2j m Now we suppose that d = m + 1 and {αj}j=1 = {α }j=0 with |α| = 1. This is in fact the case when we consider the m-symmetric power of an L-function associated with a normalized newform for SL(2, Z), (1 − αp−s)(1 − αp¯ −s) being the decomposition of its p-th Euler factor and α = α(p) is normalized by |α| = 1. In our case, (1.10) becomes mN (m−2)N mN α α ··· α¯ m(m−1) (m−2)(m−1) m(m−1) α α ··· α¯ m(m−2) (m−2)(m−2) m(m−2) α α ··· α¯ Dm+1(N) = ...... αm αm−2 ··· α¯m

1 1 ··· 1 We have

d m−2j m Proposition 1.3.1. Assume that d = m + 1 and {αj}j=1 = {α }j=0 with some |α| = 1. Then m−1 m−1 Y j j m−jY N−j N−j  (1.11) Dm+1(N) = (α − α¯ ) (α − α¯ ) . j=1 j=0 Moreover, setting α =: eiθ, we get m Y sin(k + j)θ (1.12) h = . k sin jθ j=1 1.3. AN INTERESTING IDENTITY 7

For m = 1, (1.12) is the well known trigonometric expression for the p-part of the coefficients of L(s, f) (see Murty [53]). The extension to general m’s appears to be new.

Proof. We compute the determinant defining Dm+1(N) with respect to its first row. The Vandermonde formula and a few of algebraic manipulations give

m−1 m j m+1−u m+1−u Y j j m−j X j (N− m−1 )(m−2j) Y α − α¯ D (N) = (α − α¯ ) (−1) α 2 . m+1 αu − α¯u j=1 j=0 u=1

Hence (1.11) follows if we prove the equality

m j m+1−u m+1−u m−1 X j (N− m−1 )(m−2j) Y α − α¯ Y N−j N−j (−1) α 2 = (α − α¯ ). αu − α¯u j=0 u=1 j=0

We assumed |α| = 1; so, by collectingα ¯ = α−1, the above identity becomes

m j m+1−u m−1 X j (N+ 1−m−j )(m−j) Y y − 1 Y N−j (1.13) (−1) y 2 = (y − 1) , yu − 1 j=0 u=1 j=0 with y = α2. This is a polynomial identity and we prove it showing the equality of the coefficients of y(m−j)N on left and right side, i.e., verifying that

j m+1−u j (m−j) 1−m−j Y y − 1 j X −(k +k +···+k ) (1.14) (−1) y 2 = (−1) y 1 2 m−j , yu − 1 u=1 0≤k1

m−j m+1−u j(j+1)−jm Y y − 1 X −(k +k +···+k ) y 2 = y 1 2 j for j = 0, . . . , m , yu − 1 u=1 0≤k1

m−j m+1−u Y y − 1 X k +k +···+k − j(j+1) (1.15) = y 1 2 j 2 for j = 0, . . . , m . yu − 1 u=1 1≤k1

Actually, (1.15) is a combinatorial identity. In fact, we consider

m−j Y ym+1−u − 1 C (y) := , j,m yu − 1 u=1 8 1. ARITHMETICAL RELATIONS COMING FROM EULER PRODUCTS

the left side of (1.15). It is easy verifying Cm−j,m = Cj,m. Moreover

j j Y ym+2−u − 1 Y ym+1−u − 1 C − C = C − C = − = j,m+1 j,m m+1−j,m+1 m−j,m yu − 1 yu − 1 u=1 u=1 j j Y 1 Y  = (ym+2−u − 1) (ym+1 − ym+1−j) = yu − 1 u=1 u=2 j j−1 Y 1 Y  = ym+1−j(yj − 1) (ym+1−u − 1) = ym+1−jC , yu − 1 j−1,m u=1 u=1 so the recurrence

m+1−j Cj,m+1(y) = Cj,m(y) + y Cj−1,m(y) j = 0, . . . , m

holds. With this notation, (1.15) becomes

j(j+1) X k1+k2+···+kj − (1.16) Cj,m(y) = y 2 ,

1≤k1

and (1.11) follows if we prove (1.16) for j = 0, . . . , m. But (1.16) is trivial for m = 1, and by induction we have

m+1−j Cj,m+1(y) = Cj,m(y) + y Cj−1,m(y) = X k +k +···+k − j(j+1) m+1−j X k +k +···+k − j(j−1) = y 1 2 j 2 + y y 1 2 j−1 2 =

1≤k1

− j(j+1)  X k +k +···+k X k +k +···+k +(m+1) = y 2 y 1 2 j + y 1 2 j−1 =

1≤k1

1≤k1

Hence (1.16) holds, and (1.11) follows. The second part of Proposition 1.3.1 follows easily from (1.10) and (1.11). 

Remark 1.3.1. The polynomials Cj,m(y) are known as Gauss polynomials and are of m
independent interest. They generalize the binomial coefficients (since Cj,m(1) = j ) and represent the number of projective subspaces of dimension j − 1 of a projective space of dimension m − 1 over a field with y elements. In this sense our proof is combinatorial. It is easy verifying that deg Cj,m(y) = j(m−j), Cj,m ∈ N[y] and that they are complete, Pj(m−j) (j,m) u (j,m) in the sense that Cj,m(y) =: u=0 gu y , with gu 6= 0. Moreover, every Cj,m(y) is j(m−j) (j,m) 2 unimodal, i.e., the sequence {gu }u=0 is non-decreasing, see von Kock [32]. 1.4. PRODUCT OF COEFFICIENTS IN DIRICHLET SERIES 9

1.4. Estimates for products of coefficients of Dirichlet series with Euler product ∞ 1.4.1. A general result. Let {hk}k=0 be a sequence with h0 = 1 and extended to k < 0 by setting hk = 0 in this range. We assume that the sequence hk satisfies the relation d X u (1.17) h1hk = hk+1 + xu hk+1−u , u=2

d where d ≥ 1 is a fixed integer, the degree of the sequence, and {xu}u=2 are non-negative real variables. The purpose of this section is to prove (1.26) below, giving an estimate for the product hlhk in terms of a linear combination of the hu’s involving the variables {xu}.

Remark 1.4.1. When d = 1 the relation (1.17) is reduced to h1hk = hk+1. We have

Proposition 1.4.1. Assume (1.17). Then there exist polynomials Pu,l ∈ N[x2, ···, xd] such that dl X (1.18) hlhk = Pu,l(x) hk+l−u for every l, k ≥ 0. u=0

Moreover, if we assume P0,0 = 1 and Pu,l = 0 in the following cases: u < 0 or l < 0 or u > dl or u = 1, such polynomials satisfy the recursion

d X g
(1.19) Pu,l+1(x) = Pu,l(x) + xg Pu−g,l(x) − Pu−g,l+1−g(x) . g=2

Proof. We introduce the variables x0 and x1, since in this way we can write h1hk = Pd u u=0 xuhk+1−u and relation (1.17) is recovered setting x0 = x1 = 0. We prove the claim recursively on l, and we first observe that it is true when l = 1, for every k. Next, we assume it for the values ≤ l and for every k, and we verify it for l + 1, for every k. We have dl dl d X X X g h1(hlhk) = Pu,l(x) h1hk+l−u = Pu,l(x) xg hk+l+1−u−g = u=0 u=0 g=0 dl d d(l+1) X X g X  X g  = xg Pu,l(x) hk+l+1−u−g = xg Pu,l(x) hk+l+1−γ = u=0 g=0 γ=0 g+u=γ 0≤g≤d d(l+1) X  X g  = hk+l+1 + xg Pu,l(x) hk+l+1−γ γ=1 g+u=γ 0≤g≤d 10 1. ARITHMETICAL RELATIONS COMING FROM EULER PRODUCTS

and d d X u X u (h1hl)hk = xu hl+1−uhk = hl+1hk + xu hl+1−uhk = u=0 u=1 d d(l+1−u) X X u = hl+1hk + xu Pρ,l+1−u(x) hk+l+1−ρ−u = u=1 ρ=0 dl X X u  = hl+1hk + xu Pρ,l+1−u(x) hk+l+1−γ . γ=1 u+ρ=γ 1≤u≤d 0≤ρ≤d(l+1−u) Comparing these identities we get d(l+1) X  X g h X g i  (1.20) hl+1hk = hk+l+1 + xg Pu,l(x) − xg Pu,l+1−g(x) δγ≤dl hk+l+1−γ , γ=1 g+u=γ g+u=γ 0≤g≤d 1≤g≤d 0≤u≤d(l+1−g)

where δγ≤dl is 1 or 0 according to γ ≤ dl or otherwise. The existence of the polynomials Pu,l+1 follows form (1.20). Moreover,

X g h X g i (1.21) Pγ,l+1(x) := xg Pu,l(x) − xg Pu,l+1−g(x) δγ≤dl = g+u=γ g+u=γ 0≤g≤d 1≤g≤d 0≤u≤d(l+1−g)

X g h X g i = Pγ,l(x) + xg Pu,l(x) − xg Pu,l+1−g(x) δγ≤dl , g+u=γ g+u=γ 2≤g≤d 2≤g≤d 0≤u≤d(l+1−g) where the assumption x1 = 0 has been introduced. By the conditions Pu,l = 0 for u < 0, or d < 0 or u > dl, (1.21) becomes (1.19). 

The particular form of (1.19) suggests the analysis of the polynomials Du,l := Pu,l − Pu,l−1; we have

Proposition 1.4.2. The polynomial sequence Du,l := Pu,l − Pu,l−1 verifies the recur- sion  D (x) = Pd xg Pg−1 D (x)  u,l+1 g=2 g ρ=1 u−g,l+1−ρ (1.22) D0,0 = 1  Du,l = 0 when u < 0, or u > dl, or l < 0 .

Moreover, Du,l ∈ N[x2, ··· , xd] and it is homogeneous of degree u in the sense that u Du,l(λx) = λ Du,l(x). Proof. The recursive relation (1.22) is deduced from (1.19). The other claim is an easy consequence of (1.22). 

We need another property of the sequence Du,l. 1.4. PRODUCT OF COEFFICIENTS IN DIRICHLET SERIES 11

Proposition 1.4.3. If l ≥ u + d, then Du,l = 0. Proof. We prove the claim recursively on u; it is true when u = 0 and u = 1 for every l > 0. We suppose the claim to hold up to u, and we prove it for u + 1. Let l ≥ u + 1 + d. By the identity d g−1 X g X Du+1,l(x) = xg Du+1−g,l−ρ(x), g=2 ρ=1 the claim is the same as proving that every Du+1−g,l−ρ is zero. The inductive hypothesis assures it if l − ρ ≥ u + 1 − g + d, and this inequality holds since we assumed l ≥ u + 1 + d and ρ < g. 

Remark 1.4.2. Let l0(u) be the smallest value such that Du,l = 0 for l ≥ l0(u); Proposition 1.4.3 states l0(u) ≤ u + d, but it is easy to verify that the inequality is not sharp. Nevertheless, finding the exact value of l0(u) is not necessary here.

We have Du,l(x) ∈ N[x], hence Pu,l1 (x) ≤ Pu,l2 (x) for l1 ≤ l2 (recall that we assume + Pl xi ∈ R ). Clearly, we have Pu,l = g=0 Du,g and hence we define ∞ X Pu(x) := Du,g(x). g=0 Observe that the sum is actually finite by Proposition 1.4.3, and the inequality

Pu,l(x) ≤ Pu(x) for every u, l, x

holds. The existence of the polynomials Pu is essential for our purposes, and we summarize their principal properties in the following proposition.

Proposition 1.4.4. There exist polynomials Pu ∈ N[x2, ··· , xd] such that + (1.23) Pu,l(x) ≤ Pu(x) ∀u, l, ∀x ∈ R and verifying the recurrence  P (x) = Pd (g − 1)xg P (x) if u ≥ 2  u g=2 g u−g (1.24) Pu = 0 if u < 0  P0 = 1, P1 = 0 .

Moreover, Pu is homogeneous of degree u and the estimate √ u (1.25) Pu,l(x) ≤ Pu(x) ≤ [ 2(x2 + ··· + xd)] holds. P∞ Proof. We have already shown that the polynomials Pu = l=0 Du,l verify (1.23). Such polynomials belong to N[x] and are homogeneous of degree u since the same prop- erties hold for every Du,l. The recursive relation (1.24) is recovered from (1.22) by the definition of Pu. 12 1. ARITHMETICAL RELATIONS COMING FROM EULER PRODUCTS

M + In order to prove (1.25), we first remark that if {yi}i=1 ∈ R and M i Xh yi i f (y) := , M PM i=1 j=1 yj

then fM (y) ≤ 1. We prove this fact inductively over M. It is true for M = 1. By the inductive hypothesis fM−1(y1, ··· , yM−1) ≤ 1, we get

M−1 i y1 yM Xh yi i 1 ≥ f (y + y , y , ··· , y ) = + + ≥ M−1 1 M 2 M−1 PM PM PM j=1 yj j=1 yj i=2 j=1 yj M−1 M i y1 h yM i Xh yi i ≥ + + = f (y , y , ··· , y ). PM PM PM M 1 2 M j=1 yj j=1 yj i=2 j=1 yj Now we can prove (1.25), once again by induction. It is true when u = 0 and it is trivial when u < 0. By the recursive law (1.24), we have d d g √ d X g hX (g − 1)xg i X u Pu(x) = (g − 1)x Pu−g(x) ≤ √ ( 2 xj) ≤ g Pd g g=2 g=2 ( 2 j=2 xj) j=2 d g √ d √ d hX xg i X u X u ≤ ( 2 xj) ≤ ( 2 xj) , Pd g g=2 ( j=2 xj) j=2 j=2

g/2 where we have used the inequality g − 1 ≤ 2 and the previous estimate fM (y) ≤ 1, and the result follows. 

Remark 1.4.3. Since every Du,l is homogeneous, Pu,l is homogeneous too (as directly verified by (1.19)). This immediately implies that there exists a constant c = c(l) such u that Pu,l(x) ≤ (c(l)(x2 + ··· + xd)) , for every u. Obtaining (1.25) is important since it shows that c(l) is in fact independent of l.

1.4.2. Estimates for coefficients. Let {hk} be defined as in (1.1). Relation (1.3) shows that h1 = s1, so (1.3) can be reformulated as d h1hk−1 = hk + s2hk−2 + ··· + (−1) sdhk−d . Comparing this relation with (1.17), by (1.18) and (1.25) we get

d min {l,k} √ d X u X 1/j (1.26) |hlhk| ≤ |hl+k| + R |hl+k−u| , with R := 2 |sj| . u=2 j=2 By multiplicativity, (1.26) easily implies the following ”global” result

P∞ −s Proposition 1.4.5. Let n=1 ann be a Dirichlet series with Euler product d Y Y −s −1 (1 − αj(p)p ) , p j=1 1.5. THE ”FUDGE FACTOR” AND RANKIN-SELBERG CONVOLUTION 13

and let R(n) be the totally multiplicative function defined by √ d X 1/j R(p) = 2 |sj(p)| for every p. j=2 Then X∗ (1.27) |a a | ≤ R(d)|a nm | for every n, m ≥ 1, n m d d|(n,m)d d|nm where P∗ denotes that the summation is restricted to square-full divisors, i.e., d = 1 and if p|d then p2|d.

Remark 1.4.4. When d = 1 the sequence {an} becomes totally multiplicative and estimate (1.27) reduces to the trivial |anam| ≤ |anm|.

Remark 1.4.5. When d = 2, it can be proved that the sequence {an} satisfies the identity X anam = R(d)a nm , d2 d|(n,m)

where R is the completely multiplicative function defined by R(p) = s2(p). Comparing this equality with (1.27), we observe that (1.27) is almost optimal. In fact, extra terms appear in (1.27), but they do not affect our application of Proposition 1.4.5 in the next chapter. The same phenomenon appears for any d ≥ 2.

1.5. The ”fudge factor” and Rankin-Selberg convolution Given two Dirichlet series with polynomial Euler product ∞ a ∞ b −1 −1 X an Y Y αj(p) X bn Y Y βj(p) L(s, A) = = 1 − , L(s, B) = = 1 − , ns ps ns ps n=1 p j=1 n=1 p j=1

Rankin, Selberg and Linnik suggested to study the sequence {anbn} by considering the P −s scalar product series n anbnn . The analytic properties of this series can be investigated relating it to the Rankin- Selberg convolution L(s, A ⊗ B), defined by a b −1 Y Y Y αi(p)βj(p) L(s, A ⊗ B) := 1 − . ps p i=1 j=1 Actually, the relation between the scalar product and the Rankin-Selberg functions is given by means of a ”fudge factor” F(s, A, B), defined by

X −s (1.28) L(s, A ⊗ B) =: F(s, A, B) anbnn . n The fudge factor and its influence on the analytic behaviour of the scalar product function has been extensively investigated by Moroz [44–52] and Kurokawa [33–38]. Roughly, they prove that if A and B come from unitary representations of groups, then F(s, A, B) 14 1. ARITHMETICAL RELATIONS COMING FROM EULER PRODUCTS

is meromorphic for σ > 0, regular and non-vanishing for σ > 1/2 but has the line σ = 0 as natural boundary if (a, b) 6= (1, d), (d, 1) or (2, 2). It follows that analytic continuation to σ > 1/2 is possible for Rankin-Selberg convolution if and only if this is possible for the scalar product, but a continuation to C is possible for both only in cases (a, b) = (1, d), (d, 1) or (2, 2). Here we present some complements to this theory. 1.5.1. Local aspect. Both the scalar product and the Rankin-Selberg convolution have an Euler product, so a representation

Y −s F(s, A, B) = Fp(p ) p holds as well, with a b ∞ −1 Y Y −1 X k Fp (x) =: fp(α, β, x) and fp(α(p), β(p), x) = (1 − αi(p)βj(p)x) apk bpk x . i=1 j=1 k=0

Therefore, we may restrict to the study of the local factors fp and, with abuse of notation about ak and bk, we define a ∞ b ∞ Y −1 X k Y −1 X k (1 − αjx) =: akx , (1 − βjx) =: bkx , j=1 k=0 j=1 k=0 and a b ∞ Y Y X k (1.29) f(α, β, x) := (1 − αiβjx) akbkx . i=1 j=1 k=0 We start giving a simple proof of the following

Proposition 1.5.1. f(α, β, x) is a polynomial of Z[α, β, x]. 2 Moreover, f ≡ 1( mod x ), and degx f(α, β, x) ≤ ab − max {a, b}. A similar result is already proved in [36] and [51], but only the weaker estimate degx f(α, β, x) ≤ ab−1 is proved there. Moreover, Proposition 1.5.1 shows that the fudge factor F(s, A, B) has a polynomial Euler product as well. Remark 1.5.1. Proposition 1.5.1 extends the well known Ramanujan identity 2 ∞ k+1 k+1 k+1 k+1 1 − α1α2β1β2x X α − α β − β = 1 2 1 2 xk . (1 − α1β1x)(1 − α1β2x)(1 − α2β1x)(1 − α2β2x) α1 − α2 β1 − β2 k=0

Remark 1.5.2. Choose α1 = ··· αa = β1 ··· βb = 1 with b ≥ a. Then we have ak = Pa(k), as in Section 1.1, and ∞ ∞ X  ∂  X f(α, β, x) = (1 − x)ab P (k)P (k)xk = (1 − x)abP x P (k)xk = a b a ∂x b k=0 k=0  ∂  1 = (1 − x)abP x . a ∂x (1 − x)b 1.5. THE ”FUDGE FACTOR” AND RANKIN-SELBERG CONVOLUTION 15

∂ n −b n −b Moreover, (x ∂x ) (1 − x) ∼ (−b) (1 − x) as x → ∞, so we get P (−b) f(α, β, x) ∼ (1 − x)ab a as x → ∞, (1 − x)b

where Pa(−b) 6= 0 since we suppose b ≥ a. In this case, therefore, degx f(α, β, x) = ab − b = ab − max {a, b}. This provides a simple argument pointing toward Proposition 1.5.1.

Proof. We treat {α} and {β} as variables, so we can assume αi 6= αj, βi 6= βj for i 6= j, recovering the result on the nodal lines αi = αj (and similar linear subspaces) by the continuity of f on Ca+b+1. Under this hypothesis (1.4) holds, and hence

a k+a−1 b k+b−1 X αj X βj a = , b = . k Q (α − α ) k Q (β − β ) j=1 i6=j i j j=1 i6=j i j Therefore

∞ a b a−1 b−1 X k X X αi βj 1 akbkx = Q Q (αu − αi) (αv − αj) 1 − αiβjx k=0 i=1 j=1 u6=i v6=j and hence a b a−1 b−1 Q X X αi βj u,v(1 − αuβvx) (1.30) f(α, β, x) = . 1 − α β x Q (α − α ) Q (β − β ) i=1 j=1 i j u6=i u i v6=j v j

Identity (1.30) shows that f(α, β, x) ∈ Z(α, β)[x]; this and equations f(α, β, x) = f(xα, β, 1) = f(α, xβ, 1) show that f(α, β, x) ∈ Z[α, β, x]. Further, it is easy to see that f = 1( mod x2).

We finally prove the upper-bound for degx f. For x → ∞, (1.30) becomes

f(α, β, x) = a b ∞ Y X X X −1 1 (1 − α β x) αa−1βb−1 ∼ u v i j αrβrxr Q (α − α ) Q (β − β ) u,v i=1 j=1 r=1 i j u6=i u i v6=j v j ∞ a a−r−1 ! b b−r−1 ! Y Y X 1 X α X βj ∼ −(−x)ab( α )( β ) i , u v xr Q (α − α ) Q (β − β ) u v r=1 i=1 u6=i u i j=1 v6=j u i and from (1.10) we know that

a a−r−1 X α Da(α, a − r − 1) i = , Q (α − α ) D (α, a − 1) i=1 u6=i u i a hence ∞ Y Y X 1 Da(α, a − r − 1) Db(β, b − r − 1) (1.31) f(α, β, x) ∼ −(−x)ab( α )( β ) . u v xr D (α, a − 1) D (β, b − 1) u v r=1 a b 16 1. ARITHMETICAL RELATIONS COMING FROM EULER PRODUCTS

By definition, Da(α, a − r − 1) = 0 for r = 1, . . . , a − 1 and Db(β, b − r − 1) = 0 for r = 1, . . . , b − 1, so (1.31) gives (1.32) f(α, β, x) = xab−max {a,b}(c(α, β) + O(x−1)) , with a suitable constant c(α, β). Our claim about degx f follows from (1.32).  1.5.2. The main property of the fudge factor. We consider the global fudge factor F(s, A, B), and we prove

θ1 Proposition 1.5.2. With the notation of this section, assume that αj(p)  p and θ2 βj  p , for every i, j. Then the Euler product defining F(s, A, B) converges for σ > 1/2+θ1+θ2, uniformly on compact sets. Hence, F(s, A, B) is a non-vanishing holomorphic function in this region.

The half-plane σ > 1/2 + θ1 + θ2 is generally larger than the half-plane where the Dirichlet series defining the Rankin-Selberg convolution and the scalar product converge. This fact justifies the name of the fudge factor. Nevertheless, as remarked earlier, it has many interesting properties. Proof. By Proposition 1.5.1 and the equations f(α, β, x)=f(xα, β, 1)=f(α, xβ, 1) we get deg f−2 2 X r f(α, β, x) = 1 + x gα,β,rx , r=0 with gα,β,r a symmetric polynomial of homogeneous degree r +2 in the α and β variables. Hence, there exist constants c1, c2 such that

r+2 1/2 1/a r+2 (1.33) |gα(p),β(p),r| ≤ c1 (|s1(α(p))| + |s2(α(p))| + · · · |sa(α(p))| ) ·

1/2 1/b r+2 θ1+θ2 r+2 (|s1(β(p))| + |s2(β(p))| + · · · |sb(β(p))| ) ≤ (c2p ) ,

where sj is the j-th elementary symmetrical polynomial. It follows that ∂f−2  −1 Y −1 −s Y 2 −2(σ−θ1−θ2) X θ1+θ2−σ r |F(A, B, s)| = | fp (α(p), β(p), p )| ≤ 1 − c2p (c2p ) , p p r=0

and the last product converges for σ − θ1 − θ2 > 1/2.  1.5.3. An inequality. We end the chapter by the following P −s Proposition 1.5.3. Let L(s, A) := n ann be a polynomial Euler product of degree ¯ P −s 2 d and L(s, A ⊗ A) := n Ann . Then An ≥ |an| for every n. The relation ∞ 2 −1 Y |α1(p)α2(p)|  X L(s, A ⊗ A¯) = 1 − |a |2n−s p2s n p n=1 holds when d = 2 by Ramanujan’s identity, see Remark 1.5.1, and assures the truth of the inequality in this case. For higher degree, the general relation (1.28) has to be 1.5. THE ”FUDGE FACTOR” AND RANKIN-SELBERG CONVOLUTION 17

considered. Moreover, in general the coefficients of the fudge factor are not positive, as the following example shows. Hence, in the general case the inequality of Proposition 1.5.3 is non-trivial. Example 1.5.1. We consider only a local component, and we choose a = b = 3, αi = βj = 1. Then the fudge factor is given by (see Remark 1.5.2)  ∂  1 f(1, 1, x) = (1 − x)9P x = (1 − x)4(1 + 4x + x2), 3 ∂x (1 − x)3 hence f −1(1, 1, x) = 1 + 9x2 − 16x3 + ··· . In general, the expansion   −1 2 2 2 3 f (α, α¯ , x) = 1 + |s1(α)| x − 2 <(s1(α)s2(α)¯s3(α)) − |s3(α)| x + ··· ,

holds, hence a necessary condition for the positivity of the coefficients is that s (α)s (α) (1.34) < 1 2 ≤ 1 s3(α) and this condition is generally not satisfied. For example, let L(s, f) be the L-function associated with a normalized holomorphic newform f for SL(2, Z). Then ∞ X an Y L(s, f) = = (1 − α(p)p−s)−1(1 − α¯(p)p−s)−1 ns n=1 p holds with |α(p)| = 1, by Deligne [11]. The function L(s, sym2 f), defined in Section 1.3, has Euler product of degree 3 Y L(s, sym2 f) = (1 − α2(p)p−s)−1(1 − p−s)−1(1 − α¯2(p)p−s)−1 , p

2 2 2 and condition (1.34) for L(s, sym f) becomes |α (p) + 1 +α ¯ (p)| ≤ 1, i.e., |ap2 | ≤ 1. This is a very strong condition about the coefficients of L(s, f), and it is not satisfied in general.

Proof. Both L(s, A) and L(s, A ⊗ A¯) have Euler product, so it is sufficient proving the claim for the local components. Hence we may reduce our problem to the following one: if d ∞ d ∞ Y −1 X h Y −1 X h f(x) := (1 − αjx) = ahx and F (x) := (1 − αiα¯jx) = Ahx , j=1 h=1 i,j=1 h=1

2 then Ah ≥ |ah| . We remark that

∞ d d ∞ ∞ ∞ X h d X αjx X X h h X h X h hahx = x f(x) = f(x) = f(x) αj x = ahx τhx , dx 1 − αjx h=0 j=1 j=1 h=1 h=0 h=1 18 1. ARITHMETICAL RELATIONS COMING FROM EULER PRODUCTS

Pd h where τh := j=1 αj , and similarly for the function F (x)

∞ d d ∞ X h d X αiα¯jx X X h h h hAhx = x F (x) = F (x) = F (x) αi α¯j x = dx 1 − αiα¯jx h=0 i,j=1 i,j=1 h=1 ∞ ∞ X h X 2 h Ahx |τh| x . h=0 h=1 Hence the following recursions hold ( ( ha = Ph τ a ∀h > 0 hA = Ph |τ |2A ∀h > 0 (1.35) h l=1 l h−l h l=1 l h−l a0 = 1 , A0 = 1 .

These relations prove that there exist polynomials Ph ∈ N[x1, ··· , xh] such that

h! ah = Ph(τ1, ··· , τh) 2 2 h! Ah = Ph(|τ1| , ··· , |τh| ) ∀h ≥ 0 . Moreover,

(1.36) Ph(1, ··· , 1) = h!,

since setting τl = 1 for every l, the recursion (1.35) is solved by Ah = 1 identically on h. 2 Therefore, proving Ah ≥ |ah| is the same as showing that

2 2 2 (1.37) h! Ph(|τ1| , ··· , |τh| ) ≥ |Ph(τ1, ··· , τh)| .

We have proved that Ph ∈ N[x1, ··· , xh], hence we can consider Ph as a sum of h! monomi- als (by (1.36)), every one with unitary coefficient. Then, the Cauchy-Schwarz inequality Ph! 2 Ph! 2 h! i=1|yi| ≥ | i=1 yi| gives (1.37).  There is also a more theoretical proof of Proposition 1.5.3, using the following formula (see Bump [5], Section 2.2).

d d0 Lemma (E. D. Littlewood). Let {αi}i=1 and {βj}j=1 be non-zero complex numbers, Qd Qd0 and define A := i=1 αi, B := j=1 βj. Then

d d ∞ Y Y −1 d −1X k1+2k2+3k3+···+(d−1)kd−1 (1 − αiβjx) = (1 − ABx ) Sk1,...,kd−1 (α)Sk1,...,kd−1 (β)x

i=1 j=1 k1,...,kd−1=0 if d = d0, and

d d0 ∞ Y Y −1 X kd k1+2k2+3k3+···+dkd (1 − αiβjx) = Sk1,...,kd−1 (α)Sk1,...,kd,1,...,1(β)A x

i=1 j=1 k1,...,kd=0 1.5. THE ”FUDGE FACTOR” AND RANKIN-SELBERG CONVOLUTION 19

if d < d0, with

d−1+kd−1+···+k1 d−1+kd−1+···+k1 α . . . α αd−1 . . . αd−1 1 d 1 d αd−2+kd−1+···+k2 . . . αd−2+kd−1+···+k2 αd−2 . . . αd−2 1 d 1 d S (α) := . . . / . . . . k1,...,kd−1 ...... 1+k 1+k α d−1 . . . α d−1 α1 . . . αd 1 d 1 ... 1 1 ... 1

The polynomials Sk1,...,kd−1 (α) are called Schur polynomials.

Remark 1.5.3. We observe that in this lemma it is not assumed that A and B be non-vanishing, so that the first equality can be deduced from the second one and vice versa if one takes into account the identity

kd−1 Sk1,...,kd−1 (0, α2, . . . , αd−1) = α2 ··· αd−1 Sk1,...,kd−2 (α2, . . . , αd−1).

0 When d = d and βi =α ¯i the former identity gives ∞ ∞ 2 −1 2 X An Y |A(p)|  X |a(m1, . . ., md−1)| = 1 − , s ds 2 d−1 s n p (m1m ··· m ) n=1 p m1,...,md−1=1 2 d−1 Qd where A(p) = i=1 αi(p) and a(m1, . . . , md−1) is multiplicative in every entry with

k1 kd−1 a(p , . . ., p ) := Sk1,...,kd−1 (α(p)) . Hence X 2 2 2 An ≥ |a(m1, . . ., md−1)| ≥ |a(n, 1, . . ., 1)| = |an| , d−1 m1···md−1=n and the proposition is again proved. We end the chapter observing that our first proof is more direct and conceptually easier.

CHAPTER 2

The Siegel zero

2.1. Introduction Let χ be a primitive Dirichlet character modulo q and ∞ X χ(n) L(s, χ) := ns n=1 be the associated Dirichlet L-function. It is well known that L(s, χ) is an entire function if χ 6= χ0, while if χ = χ0 there is a simple pole at s = 1. Moreover, L(s, χ) satisfies the functional equation

1−s+νχ √ s+νχ  q  2 1 − s + ν  q  q  2 s + ν  Γ χ L(1 − s, χ) = i−νχ Γ χ L(s, χ¯), π 2 τ(χ) π 2

νχ where νχ ∈ {0, 1} is defined by χ(−1) = (−1) and τ(χ) is the Gauss sum (see Daven- port [10]). The arithmetical interest of these functions lies in the Euler product representation Y χ(p)−1 L(s, χ) = 1 − for σ > 1, ps p which connects such functions with prime numbers. In fact, the Dirichlet L-functions have been employed to show the existence of infinitely many primes in every arithmetic progression a + qn when (a, q) = 1. This follows from the fact that L(1, χ) 6= 0. The non-vanishing of L(s, χ) on the line σ = 1 implies the stronger result 1 x (2.1) #{p : p < x, p ≡ a(q)} ∼ if (a, q) = 1 . φ(q) ln x A quantitative version of (2.1) is possible as soon as a larger zero-free region in the strip 1/2 < σ ≤ 1 is available, and an almost optimal result can be obtained from the Generalized Riemann Hypothesis L(s, χ) 6= 0 for σ > 1/2 and every χ, which implies 1 x #{p : p < x, p ≡ a(q)} = + O (x1/2+) if (a, q) = 1 . φ(q) ln x q It is therefore clear that proving a zero-free region, as wide as possible, for the L- functions is a very interesting problem with many important implications. In particular, the following aspects of the problems are important: 21 22 2. THE SIEGEL ZERO

∗ uniformity with respect to the modulus q, ∗ non-existence of zeros close to s = 1. We briefly review the basic results on zero-free regions for Dirichlet L-functions. We refer to Davenport [10] for the proofs. Theorem (de la Vall`ee-Poussin). Let L(c, q) be the region {s = σ + it : σ > 1 − c ln q(|t|+2) }. There exists an absolute and effective constant c > 0 such that ∗ if χ is a complex character (χ 6=χ ¯), then L(s, χ) 6= 0 in L(c, q), ∗ if χ is a real character (χ =χ ¯), then L(s, χ) 6= 0 in L(c, q), except for at most one zero β˜. If β˜ exists, it is real and simple. We remark that the different behaviour for complex and real characters is due to the 2 2 fact that χ is induced by χ0 if χ is real, and hence in this case L(s, χ ) has a simple pole at s = 1. The above zero-free region is not the best known at present. Nevertheless, such result is obtained by a very general technique and can be generalized to wider classes of L- functions (see the following Proposition 2.2.1). Definition. The zero β˜ is called Siegel zero. There are several results, showing that the existence of the Siegel zero should be an exceptional event. For example, we have Theorem (Landau-Page). There exists an effective positive constant c > 0 such that for every z ≥ 3, there is at most one real primitive χ to a modulus q ≤ z for which L(s, χ) has a zero β˜ satisfying c 1 − β˜ < . ln z From this short discussion it is clear that we would like to prove that the Siegel zero does not exist. This conjecture is supported not only by the Generalized Riemann Hypothesis, but also by the recent results of Goldfeld, Hoffstein and Lieman [20], Hoffstein and Ramakrishnan [25] and Banks [1] showing that β˜ does not exist for a certain class of L-functions, roughly the class of L-functions associated with unitary cuspidal automorphic representations of GL(2) and GL(3). Moreover, [25] shows that the Siegel zero β˜ does not exist for every GL(r), r ≥ 2, if the following conjecture holds (see Subsection 2.3.4). Conjecture (Langlands [39]). For every cusp form π on GL(r) and π0 on GL(r0), 0 0 there exists an automorphic representation π  π of GL(rr ) whose L-function is the Rankin-Selberg convolution of L(s, π) and L(s, π0). Nevertheless, the case r = 1 corresponding to the classical L(s, χ) functions appears to be of greater difficulty. Therefore, it is very interesting trying to establish how close to s = 1 can the Siegel zero be. A simple argument gives an upper-bound for β˜ in terms of an upper-bound for L0(s, χ) in a neighborhood of s = 1 and a lower-bound for L(1, χ). In fact, L(1, χ) = L(1, χ) − L(β,˜ χ) = L0(η, χ)(1 − β˜) for some η ∈ [1 − β,˜ 1], 2.1. INTRODUCTION 23

hence, since (2.2) L0(s, χ)  ln2 q for |1 − s|  1/ ln q , we have (2.3) 1 − β˜  L(1, χ) ln−2 q . We recall that Hecke proved a lower-bound for L(1, χ) in terms of β˜.

Theorem (Hecke). Let χ be a real character modulo q that is not induced by χ0, and let 0 < β  1/ ln q be such that L(s, χ) 6= 0 in [1 − β, 1]. Then there exists an effective constant c > 0 such that (2.4) L(1, χ) ≥ cβ . Hence, localizing β˜ and obtaining a lower-bound for L(1, χ) are essentially equivalent problems. An easy estimate follows from the Dirichlet class number formula πh(−D) (2.5) L(1, χD) = √ , D where D√ > 3 is a fundamental discriminant and h(−D) the class number of the quadratic −1/2 field Q[ −D]. Since h(−D) ≥ 1, (2.5) implies that L(1, χD)  D , hence √ 1 − β˜  1/ D log2 D . Any better lower-bound is considered a non-trivial result. One of the most interesting results in this subject is Siegel’s theorem, which can be formulated in two equivalent ways by (2.3) and (2.4). Theorem (Siegel). For every  > 0 there exists a constant () > 0 such that 1 − β˜ ≥ c()q− for every primitive character modulo q. Theorem (Siegel). For every  > 0 there exists a constant c() > 0 such that L(1, χ) ≥ c()q− for every primitive character modulo q. This theorem is an important advance in number theory, having among its immediate consequences the Gauss conjecture about the class number of imaginary quadratic fields, i.e., h(−D) → ∞ as D → +∞. Unfortunately, there is a serious drawback in this theorem: the constant c() is ineffec- tive. This affects its corollaries: for example, it does not permit to find all the imaginary quadratic fields with fixed class number h. However, this problem has been solved for h = 1, 2 by Baker [2, 3] and Stark [61, 62], and, at least theoretically, for any value of h by Gross and Zagier [23]. In fact, as a consequence of an important work on the Birch and Swinnerton-Dyer conjectures, Gross and Zagier have shown that Goldfeld’s approach [19] leads to an effective lower bound for h(−D) (see also the survey of Zagier [64]). A theorem of Tatuzawa partially overcomes the drawback of Siegel’s theorem: it states that there exists an effective value for c() such that Siegel’s lower bound holds with an exception at most. 24 2. THE SIEGEL ZERO

Theorem (Tatuzawa). For every  > 0 there exists an effective constant c() such that L(σ, χ) 6= 0 for σ ∈ [1 − c()q−, 1] or, equivalently, L(1, χ) ≥ c()q− , hold with an exception at most. The original proof of Siegel’s theorem has been considerably simplified. We present here two simple versions of such proof, for future reference.

proof of Siegel’s theorem. Let χ, χ0 be different, primitive and non-principal real characters modulo q, q0 respectively. We define the function

F (s) := ζ(s)L(s, χ)L(s, χ0)L(s, χχ0) = Y 1 −1 χ(p)−1 χ0(p)−1 χ(p)χ0(p)−1 1 − 1 − 1 − 1 − . ps ps ps ps p P∞ −s Considering the logarithm of F (s), it is easy to see that F (s) = n=1 ann for σ > 1, with a0 = 1 and an ≥ 0 for any n. Moreover, F (s) has a simple pole at s = 1 with 0 0 Ress=1 F (s) = L(1, χ)L(1, χ )L(1, χχ ), and it is regular in C\{1}. The fundamental step is proving an inequality of the form

0 −c(1−σ) (2.6) Ress=1 F (s)  (1/2 − F (σ))(1 − σ)(qq ) for 2/3 < σ < 1 , uniformly over q and q0, where c > 0 is some fixed constant. We present below two substantially equivalent ways to prove (2.6). Now, we fix  > 0. Siegel’s theorem is trivially true if there is no modulus q and real character χ ( mod q) such that L(s, χ) has a zero in [1 − , 1], so we may assume that 0 0 such data exist. Let β be a zero in [1 − , 1] of L(s, χ). We choose q = q, χ = χ, σ = β in (2.6), thus obtaining, since F (β) = 0, that −c Ress=1 F (s)  (1 − β)(qq) .

Observe that both β and q depend on  in an ineffective way. We therefore get −c (2.7) L(1, χ)L(1, χ)L(1, χχ)  q , uniformly over q. But for every non-principal character η modulo q the upper-bound (2.8) L(1, η)  ln q holds, hence (2.7) gives

−(c+1) L(1, χ)  q uniformly over q, and the theorem is proved. Now we prove (2.6). 2.1. INTRODUCTION 25

Estermann’s approach (ch. 21 of [10]). By the regularity of F (s), the equality P∞ n n (n) F (s) = n=0 bn(2 − s) holds in |s − 2| < 1, with bn = (−1) F (2)/n! ≥ 0 and b0 > 1. Moreover, ∞ Ress=1 F (s) X F (s) − = (b − Res F (s))(2 − s)n s − 1 n s=1 n=0 converges in |s − 2| < 2 since the left-side is regular there. On the circle |s − 2| = 3/2 0 c1 we have F (s)  (qq ) for some positive constant c1 by the Phragm´en-Lindel¨oftheorem. Hence Cauchy’s formula gives

0 c1 n |bn − Ress=1 F (s)|  (qq ) (2/3) for every n. 2−σ 0 We choose σ such that | 3/2 | < 1 and assume U = U(q, q ) > 0 such that ∞ ∞ X X2 − σ n |b − Res F (s)|(2 − σ)n  (qq0)c1 < 1/2 , n s=1 3/2 n=U n=U hence

U−1 Ress=1 F (s) X F (σ) − ≥ b − 1/2 − Res F (s) (2 − σ)n ≥ σ − 1 0 s=1 n=0 (2 − σ)U − 1 1/2 − Res F (s) . s=1 1 − σ Therefore (2 − σ)U (2.9) F (σ) ≥ 1/2 − Res F (s) s=1 1 − σ

c1 0 0 for 1/2 < σ < 1 and U > 3/2 ln qq . For 2/3 < σ < 1 we can take U  ln qq , so ln 2−σ

0 (2 − σ)U < ec2 ln qq ln(1+(1−σ)) < (qq0)c2(1−σ)

for some positive constant c2, and (2.9) gives (2.6).

Chowla-Goldfeld’s approach [18]. Since F (s) has polynomial behaviour on ver- tical strips, while Γ(s) decays exponentially, the equality

∞ X 1 Z 2+i∞ e−1/x ≤ a n−σe−n/x = F (s + σ)Γ(s)xsds = n 2πi n=1 2−i∞ Z −1+i∞ 1−σ 1 s Ress=1 F (s) Γ(1 − σ)x + F (σ) + F (s + σ)Γ(s)x ds 2πi −1−i∞ holds by the residue theorem for x > 1 and 1/2 < σ < 1. By standard estimates on vertical strips for L(s, χ), we get (qq0)c  e−1/x ≤ Res F (s) Γ(1 − σ)x1−σ + F (σ) + O , s=1 x 26 2. THE SIEGEL ZERO

0 c 0 c3 (qq )
−1/x and choosing x = (qq ) for a suitable c3 > 0 we have O x < e − 1/2. Hence we get c3(1−σ) 1/2 − F (σ)  Ress=1 F (s)(q1q2) Γ(1 − σ), implying (2.6).  In the following sections we will study axiomatically defined classes of L-functions containing most functions of number-theoretical interest, and we will prove Siegel-type theorems for these classes.

2.2. The axiomatic L∗-classes We say that a set of L-functions is an L∗-class if the following axioms are satisfied:

(A0) (Euler product) There exists a positive integer dA and a sequence of complex square matrices {Ap} of order dA such that

dA ∞ Y Y Y X an L(s, A) := det(1 − A p−s)−1 =: (1 − α (p)p−s)−1 := p j ns p p j=1 n=1 is absolutely convergent for σ > 1. (A1) (Meromorphic continuation) The function L(s, A) has analytic continuation over C except possibly at s = 1, where it has a pole of order mA ≥ 0. (A2) (Growth condition) There exists an absolute constant 0 < δ < 1/2 such that for every  > 0

L(s, A) A, exp(exp(|t|)) , |t| → ∞ , uniformly for −δ ≤ σ ≤ 1 + δ, |t| > 1. ∗ (A3) (Functional equation) There exists a sequence {Ap} of complex square matrices satisfying (A1), (A2), (A3) and such that ∗ φ(s, A) = ωAφ(1 − s, A ), where N s Y φ(s, A) := QA Γ(λjs + µj)L(s, A)(2.10) j=1 N ∗ s Y ∗ φ(s, A ) := QA Γ(λjs +µ ¯j)L(s, A ), j=1

with QA > 0, λj > 0, <µj ≥ 0 and |ωA| = 1.

Definition 2.2.1. QA is the main parameter of L(s, A). By abuse of no- tation we denote max{2,QA} again by QA. Moreover, we call weight of L(s, A) the quantity N N X X WA := 1 + | (2<µj − 1)| + |=µj| . j=1 j=1 2.2. THE AXIOMATIC L∗-CLASSES 27

(A4) (Growth condition on the coefficients) There exists 0 ≤ θ < 1 such that |αj(p)|  θ p for every prime p, uniformly over QA,WA. Moreover, the estimate 1/j 1/2 (2.11) |sj(p)|  p for every 2 ≤ j ≤ dA,

uniformly over QA and WA, holds for the elementary symmetric polynomials of α1(p),..., αd(p). (A5) (Tensor product) For any couple A, B as in (A0),. . . ,(A4), there exists a finite, possibly empty, set of primes PA,B such that

dA dB Y −s −1 Y Y Y −s −1 L(s, A ⊗ B) := det(1 − Ap ⊗ Bpp ) = (1 − αi(p)βj(p)p ) , p p i=1 j=1

dAdB Y Y −s P (s, A ⊗ B) := (1 − γi(p)p ),

p∈PA,B i=1 L(s, A ⊗ B) =: P (s, A ⊗ B)L(s, A^ ⊗ B), θ with γi(p) ∈ C, |γi(p)|  p uniformly over QA, QB, WA and WB, where θ is the constant of Axiom (A4). Moreover, we assume that L(s, A^ ⊗ B) satis- fies (A0),. . . ,(A3). By abuse of notation, we denote by QA⊗B and WA⊗B the parameters of L(s, A^ ⊗ B). At last, we assume the following compatibility conditions

ln QA⊗B  ln QA + ln QB ,(2.12)

WA⊗B  WA + WB ,(2.13) X
ln p  ln QA⊗BWA⊗B .(2.14)

p∈PA⊗B

Definition 2.2.2. The finite set PA,B is the exceptional set for the couple A, B, and P (s, A ⊗ B) is the exceptional part of L(s, A ⊗ B).

Definition 2.2.3. The parameter dA is called the (arithmetic) degree of L(s, A), and PN dA := 2 i=1 λi the (analytic) degree. Definition 2.2.4. The tensor product with a Dirichlet character χ is usually called twist, and alternative notation is

Lχ(s, A) := L(s, A ⊗ χ) := L(s, Aχ). ∗ We say that an L -class is twist closed if for every A there exists a positive integer qA such that Lχ(s, A) satisfies Axioms (A1)-(A3) for every primitive character χ modulo q with q > qA. Defining the twist only by characters with sufficiently large modulus, we skip the problems related with the existence of characters χ such that the twist Lχ(s, A) does not satisfy a functional equation of the required form (see the Dirichlet L-functions, for example). Our definition of closeness by twists is rough, but it is convenient for our needs, 28 2. THE SIEGEL ZERO

since we basically consider asymptotic results where the modulus q goes to infinity (see Section 2.3). Remark 2.2.1. For sake of simplicity, only the uniformity over the main parameter QA and the weight WA will be considered in our results. This means that the constants appearing explicitly or in symbols O,  and ∼ may depend on all parameters of the involved L-functions, but are independent on QA and WA. This does not affect the strength of the result, since in applications the interesting quantities are connected with QA and WA. Nevertheless, if necessary, it can be proved that such constants essentially depend only on the maximal degree max{dA, dA}, on the minimal number of Γ-factors in the functional equation and on the order of pole mA. Remark 2.2.2. The first part of Axiom (A4) is equivalent to the assumption that θ+ the local components of L(s, A) are convergent for σ > θ, and implies that an  n for every positive . In fact, it is possible to prove the equivalence of the following three conditions

dA mθ −) apm  m p for every m ≥ 0 1/j θ (2.15) −) |sj(p)|  p for j = 1, . . ., dA θ −) |αj(p)|  p for j = 1, . . ., dA . d m/2 Hence, Axiom (A4) is satisfied if the estimate apm  m p holds, but (A4) is a weaker condition since nothing is assumed about the first symmetric polynomial. The estimate on the elementary symmetric polynomials is remarkably weaker than the usual Ramanujan hypothesis

|αj(p)| ≤ 1 for j = 1, ··· , dA and for every prime p. This hypothesis is probably verified by L-functions of number-theoretical interest, but it 1/2 has not been proved in general. In many cases the estimate |αp|  p is the best known estimate, so that stating our results assuming only this hypothesis has obvious advantages in the applications. Examples. Let α(p) and α−1(p) be the coefficients of the Euler product of the L- function associated with a Maass form f. Then s2(p) = 1, hence the Maass forms verify (A4). Moreover, Shahidi [59] proved that p−1/5 < |α(p)| < p1/5, so that the Rankin- Selberg convolution L(s, f ⊗ g) of two Maass forms satisfies (A4). Let us consider L(s, sym2 f), the square-symmetric function associated with f. It 2 −2 has Euler product of degree 3 and α (p), 1, α (p) as coefficients. Hence s3(p) = 1 and 2 −2 1/2 2 |s2(p)| = |α (p) + 1 + α (p)| ≤ 3p , and (A4) is satisfied for L(s, sym f) as well. Maass functions are examples of automorphic L-functions, a general and extremely interesting class of functions. In this theory, an L-function, L(s, π), is associated with

every irreducible automorphic representation π of GL(d, AK), where AK is the Ad`ele ring of a global field K. It is well known that these functions satisfy Axioms (A0)-(A5) thanks to the work of many authors; we will come back to this point in Section 2.3. Here we only remark that the best general estimate about the local coefficients of such functions is 1/2 |αj(p)| < p for any j = 1,..., d , 2.2. THE AXIOMATIC L∗-CLASSES 29

(when π is unitary and unramified, see for example Gelbart and Shahidi [17], Section 2.3, or Jacquet and Shalika [28], Corollary 2.5), i.e., exactly the weakest condition under which Axiom (A4) still holds. Remark 2.2.3. In Kaczorowski and Perelli [30], Theorem 1, it is proved that in the Selberg class (see Chapter 4) the only way to produce different functional equations for the same L-function is via the Legendre-Gauss multiplication formula and the factorial formula, i.e., the identities m−1 s−1/2 1−m Y s + k  (2.16) Γ(s) = m (2π) 2 Γ m = 2, 3,..., m k=0 Γ(s + 1) = sΓ(s) .(2.17) Due to the similarity of the L∗-class and Selberg class, it is clear that the same result ∗ also holds for every L -class. It follows that, although the parameters QA, N, λi, µi, the data of L(s, A), are not uniquely defined by L(s, A), there exist combinations of the data that depend only on L(s, A). These combinations are called invariants. In particular, P it is well known that dA = 2 j λj is an invariant, and [30] proves that, for instance, 2λ dA 2 Q j P qA := (2π) QA j λj (the modulus) and ηA := j(2<µj − 1) (the parity) are also PN invariants. Following the terminology of that paper, it is easy verifying that j=1|=µj| is a reduced parameter, so that it is also invariant by Theorem 2 of [30]. Moreover, since P WA = 1 + |ηA| + j|=µj|, WA is an invariant as well.

The main result of this chapter will be the proof of inequalities of the form
−
 (2.18) QAWA  L(1, A)  QAWA .

It is clear that QA is not an invariant, but the relation among parameters QA coming from different functional equations is sufficiently good for our purpose, as we now show. We first observe that N d d  dA  A Y 2λ dA  A (2.19) ≤ λ j ≤ . 2N j 2 j=1

Hence, fixing a functional equation for L(s, A) with data Q1, N1, λj,1, µj,1, for every other functional equation with data Q2, N2, λj,2, µj,2 we have N 2λ q (2π)dA Q2 Q 1 λ j,1 1 = A = 1 j=1 j,1 , N 2λj,2 qA dA 2 Q 2 (2π) Q2 j=1 λj,2 so that

2 2 −dA (2.20) Q2 ≥ Q1N1 . It follows that uniformity in the Q aspect of estimate (2.18) is preserved, independently of the functional equation at hand. 30 2. THE SIEGEL ZERO

P Remark 2.2.4. In [30] it is proved that ξA := j(2µj − 1) is an invariant. We prefer WA to ξA for our results since there are important L-functions for which important data are contained in =µj, but they disappear in ξA. For example, consider a cuspidal Maass 1 2 form for the congruence group Γ0(N), with 4 + r , r real, as eigenvalue of the hyperbolic laplacian. Such function satisfies the functional equation √ √  N s s + ir s − ir  N 1−s 1 − s + ir 1 − s − ir Γ Γ L(s, A) = Γ Γ L(1−s, A). π 2 2 π 2 2 √ 2 Here N and r are relevant parameters, and we have QA = N/π, qA = 2N, WA = 3+2|r| and ξA = −2. Hence the r-parameter would be lost if ξA is used instead of WA.

Remark 2.2.5. Equality dA = dA holds for all known L-functions, so it is likely that it holds in general, but this is not assumed here. This conjecture is a strengthening of the degree conjecture, see [31].

Remark 2.2.6. By conjugation we have that L(s, A) and L(s, A¯) belong to the same L∗-class, so that by Axiom (A5) both L(s, A ⊗ A) and L(s, A ⊗ A¯) have good analytic properties (Axioms (A0)-(A3)). We remark that L(s, A ⊗ A¯) is a Dirichlet series with positive coefficients, hence it has a singularity on its convergence line by Landau’s theorem. Therefore, by Axiom (A1) we get that mA⊗A¯ ≥ 1. Axioms defining L∗-classes are very similar to those considered in Carletti, Monti- Bragadin and Perelli [7], the differences being that we assume a much weaker form of the Ramanujan hypothesis and a stronger form of Axiom (A5); see remark 2) of sec. 7 of [7]. In the next sections we review the basic properties of L-functions belonging to a L∗- class. The treatment follows that of [7]. 2.2.1. General lemmas. Here we present some lemmas on the basic analytic prop- erties of L-functions of an L∗-class. The results are standard, even if the dependence of the constants on QA and WA is taken into account. We recall that the constants may depend on the parameters of L(s, A), but are uniform on QA and WA (see Remark 2.2.1). We write the functional equation of L(s, A) as L(1 − s, A) = X(s)L(s, A∗), hence an easy consequence of the Stirling formula for the Γ function is Lemma 2.2.1. Let L(s, A) be in an L∗-class. Then there exists a positive constant c such that for σ > 1 and |t| > 1 we have  W c  |X(s)|  Q2σ−1|t|dA(σ−1/2) 1 + O A . σ A σ |t| Lemma 2.2.2. Let L(s, A) and L(s, B) be in an L∗-class and a, b ∈ R, a < b. There exist constants c1 = c1(a, b) > 0 and c2 = c2(a, b) > 0 such that c mA
1 (s − 1) L(s, A) a,b QAWA(|t| + 1) , c mA⊗B
2 (s − 1) L(s, A ⊗ B) a,b QA⊗BWA⊗B(|t| + 1) , 2.2. THE AXIOMATIC L∗-CLASSES 31

uniformly on the vertical strip a < σ < b. Remark 2.2.7. By Cauchy’s theorem about the derivative of a holomorphic function, this lemma implies that

(j) j! c (s − 1)mA L(s, A)
 Q W (|t| + 1)
1 , a,b 2j A A (j) j! c (s − 1)mA⊗B L(s, A ⊗ B)
 Q W (|t| + 1)
2 , a,b 2j A⊗B A⊗B

uniformly on j, where c1 and c2 are the constants in Lemma 2.2.2.

Proof. The function (s − 1)mA L(s, A) is entire by Axiom (A1) and of order 1 by Axiom (A2), the well known properties of Γ function and the functional equation of Axiom (A3). The claim about L(s, A) hence follows from the Phragm´en-Lindel¨oftheorem and Lemma 2.2.1. In a similar way we get that there exists c3 = c3(a, b) > 0 such that

c mA⊗B
3 (s − 1) L(s, A^ ⊗ B) a,b QA⊗BWA⊗B(|t| + 1) . Moreover, P (s, A ⊗ B) is bounded on vertical lines, so that the claim about L(s, A ⊗ B) follows if we prove that

c4 P (s, A ⊗ B) a,b (QA⊗BWA⊗B)

for some c4 = c4(a, b) > 0. By Axiom (A5) we have that Y |P (s, A ⊗ B)|  (1 + pθ−σ)dAdB ,

p∈PA,B hence

P θ−σ P dAdB p∈P ln(1+p ) |θ−σ|dAdB p∈P ln p c4 |P (s, A ⊗ B)|  e A,B ≤ e A,B  (QA⊗BWA⊗B) by assumption (2.14).  Let now N(T, A) := #{ρ = β + iγ : L(ρ, A) = 0, β ≥ 0, |γ| ≤ T } . As in Lemma 3 of [7], an asymptotic formula for N(T, A) can be proved. Nevertheless, we need only the following weaker estimate that is an immediate generalization of Theorem 9.2 of Titchmarsh [63]. Lemma 2.2.3. Let L(s, A) be in an L∗-class, then
N(T + 1, A) − N(T, A) = O ln(QAWA(|T | + 2)) . From Lemma 2.2.3 and Lemma α of Titchmarsh we easily get the following last lemma providing a finite version of the Weierstrass representation of L(s, A) as product over its zeros. 32 2. THE SIEGEL ZERO

Lemma 2.2.4. Let L(s, A) be in an L∗-class. Then 0 L mA X 1 − (s, A) = − + Olog(Q W (|t| + 2))
, L s − 1 s − ρ A A |t−γ|≤1 where ρ = β+iγ runs over the zeros of L(s, A), uniformly on the vertical strip −1 < σ < 2. 2.2.2. Zero-free region. We reproduce for L∗ classes some further definitions intro- duced in [7]. As usual, let L(s, A) be an L-function in an L∗-class. Definition 2.2.5. We say that L(s, A) is (analytically) irreducible if the pole of L(s, A ⊗ A¯) at s = 1 is simple. Remark 2.2.8. If L(s, A) is an irreducible element of a twist closed L∗-class, then Lχ(s, A) is irreducible as well, since L(s, Aχ ⊗ Aχ) = L(s, A ⊗ A¯)P (s, χ) with P (s, χ) a suitable Dirichlet polynomial.

Axiom (A1) implies that there exists a sequence Λn ∈ C such that 0 ∞ L X Λn(A) − (s, A) = for σ > 1. L ns n=1 Definition 2.2.6. Let L(s, A) be irreducible. Then (i) L(s, A) is complex if L(s, A ⊗ A) is holomorphic at s=1. (ii) L(s, A) is real if L(s, A ⊗ A) has a simple pole at s=1. (iii) L(s, A) is totally real if det(1 − Apx) ∈ R[x] for every prime p. (iv) L(s, A) is positive if Λn(A) ≥ 0 for every n. Observe that if A and B are in an L∗-class, then we can consider the direct sum A+B, getting the product L(s, A + B) := L(s, A)L(s, B). This show that any L∗-class is closed under the direct sum. This fact permits to give an arithmetical definition of irreducibility: Definition 2.2.7. L(s, A) is (arithmetically) reducible (or non-primitive, according to Selberg [56]) if L(s, A) = L(s, B + C) with A, B and C in the same L∗-class and B, C= 6 0. These two notions of reducibility should be equivalent, and actually the arrow arith- metically reducible =⇒ analytically reducible is heuristically trivial. In fact, A arithmeti- cally reducible implies A = B + C so that L(s, A ⊗ A¯) = L(s, (B + C) ⊗ (B + C)) = L(s, B ⊗ B¯)L(s, C ⊗ C¯)L(s, B ⊗ C¯)L(s, C ⊗ B¯) ¯ and it is expected that L(1, A ⊗ C) 6= 0, so mA⊗A¯ ≥ 2. The converse implication is proved for some L∗-class (the automorphic L-functions, for example: Moeglin and Wald- spurger [41] have proved that if π is an (arithmetically) irreducible automorphic repre- sentation, then mπ⊗π¯ = 1), but in general it is only a conjecture. For further comments about this subject see [7]. As announced in the introduction, a general zero-free region holds for L∗-classes. Proposition 2.2.1. Let L(s, A) be in an L∗-class. 2.2. THE AXIOMATIC L∗-CLASSES 33

(i) There exists a positive constant c1 such that if A is positive and mA = 1, then L(s, A) 6= 0 in the region c1 σ > 1 −
, t ∈ R , log QAWA(|t| + 2)

except for a possible simple real zero βA < 1. (ii) There exists a positive constant c1 such that if A is irreducible, complex and mA = 0, then L(s, A) 6= 0 in the region c1 σ > 1 −
, t ∈ R . log QAWA(|t| + 2)

(iii) There exists a positive constant c1 such that if A is irreducible, real and mA = 0, then L(s, A) 6= 0 in the region c1 σ > 1 −
, t ∈ R , log QAWA(|t| + 2)

except for a possible simple zero ρA = βA + iγA satisfying

20c1 βA < 1, |γA| <
. ln QAWA

If A is totally real, the zero ρA, if it exists, is real.

Definition 2.2.8. The possible zero βA (resp. ρA) is called the Siegel zero. We introduce the quantity δA by ( 1 if the Siegel zero exists δ = A 0 otherwise. Proposition 2.2.1 is very similar in its conclusions to Theorem 1 of [7], but it also takes into account the dependence on the weight. Proof. The proof follows the same ideas of Theorem 1 of [7], so we only give some hints of case (ii). In this case the inequality 2 + η2 + 4η cos φ + η2 cos 2φ ≥ 0 is employed, getting ζ0 L0 L0 L0 −2 (σ) − (σ, A ⊗ A¯) − 4< (σ + it, A) − < (σ + 2it, A ⊗ A) ≥ 0 for σ > 1. ζ L L L We observe that L0 P 0 L0 − (s, A ⊗ A) = − (s, A ⊗ A) − (s, A^ ⊗ A), L P L

so that by the assumptions in (A5) we can choose σ0, θ < σ0 < 1 such that for σ > σ0

∞ d2 P 0 X X log p XA − (s, A ⊗ A)  λ (p)m  P pmσ j p∈PA,A m=1 j=1 ∞ 2 X X m(θ−σ) X
dA log p p  log p  log QA⊗AWA⊗A

p∈PA,A m=1 p∈PA,A 34 2. THE SIEGEL ZERO

¯ uniformly on QA,WA. A similar result holds for A ⊗ A as well. It follows that there exists a segment [σ0, 1] where a result analogous to Lemma 2.2.4 holds for L(s, A ⊗ A) and L(s, A ⊗ A¯). Hence we can pursue the proof exactly as in Theorem 1 of [7], case (ii).  2.2.3. Estimates at s = 1. The aim of this subsection is to prove an upper bound for L(1, A), and more generally for L(j)(s, A), of type (j)  (2.21) L (s, A) ,j (QAWA) for |s − 1|  1/ ln(QAWA). Such estimates are an essential tool in our approach to a general Siegel-type theorem. Obtaining these estimates is quite easy when the Ramanujan hypothesis is assumed, and under this hypothesis better estimates can be proved (see Theorem 2 of [7]). When the Ramanujan hypothesis is relaxed, as in our setting, proving such estimates becomes an interesting non-trivial problem. This has been done first by Iwaniec in [27] for the Maass forms (dA = 2). Iwaniec considers only uniformity in the parameter QA. His approach is essentially as follows. A preliminary estimate X |an| S(x) :=  Qc x n  A n≤x with some constant c and for every  > 0 can be proved in a standard way. Then, the multiplicative properties of the coefficients an can be employed to obtain an estimate for S(x)2 in terms of S(x2). By iterating this relation the fundamental estimate  S(x)  (QAx) is deduced, and the claim easily follows. This method has also been used by Hoffstein-Lockhart [24] to get an analogous esti- 2 mate in the case of the symmetric square L-functions L(s, sym f)(dA = 3) associated with a Mass form f. In [24, 27] the multiplicative properties are explicitly manipulated to produce the 2 2 estimate of S(x) in terms of S(x ). This is possible since in their situation (dA = 2, 3) the multiplicative law is quite simple. When dA increases, such multiplicative law becomes more and more involved, so that producing the necessary tools to extend Iwaniec’s idea to any degree dA becomes an interesting arithmetical problem. This has been done in Section 1.4, and here we apply our previous results to get bounds of type (2.21). We also remark the following fact: a polynomial Euler product is not necessary to establish estimates of type (2.21) when the Ramanujan hypothesis is assumed, but it becomes an essential ingredient when Ramanujan hypothesis is relaxed. Theorem 2.2.1. Let L(s, A) be in an L∗-class. (i) Let f(s) :=(s − 1)mA L(s, A). Then

(j) jh  j j! i (2.22) f (s)  c (j + 1)(QAWA) ln (QAWA) + j 2 QAWA

for |s − 1|  1/ ln(QAWA) and uniformly on j, where c is a suitable positive constant. 2.2. THE AXIOMATIC L∗-CLASSES 35

(ii) If L(s, A) is irreducible and L(s, A ⊗ A¯) satisfies Axiom (A4), let f(s) := (s − 1)L(s, A ⊗ A¯). Then

(j) jh  j j! i (2.23) f (s)  c (j + 1)(QAWA) ln (QAWA) + j 2 QAWA

for |s − 1|  1/ ln(QAWA) and uniformly on j, where c is a suitable positive constant. (iii) Let the L∗-class twist closed by Dirichlet characters χ and assume that L(s, A⊗A¯) mA⊗A¯χ ¯ satisfies Axiom (A4). Let fχ(s) := (s − 1) Lχ(s, A ⊗ A), with χ a character modulo q > qA. Then

(j) jh  j j! i (2.24) f (s)  c (j + 1)(qQAWA) ln (qQAWA) + j 2 qQAWA

for |s − 1|  1/ ln(qQAWA) and uniformly on j, where c is a suitable positive constant. Here we have assumed the simplifying hypotheses

ln QAχ  ln QA + ln q , WAχ  WA , similar to (2.12) and (2.13). (iv) Suppose that L(s, A ⊗ A¯) and L(s, B ⊗ B¯) satisfy Axiom (A4) and that

d d YA YB |αi(p)| , |βj(p)| ≤ 1 , i=1 j=1

mA⊗B with dA > 1 if dA = dB. Let f(s) := (s − 1) L(s, A ⊗ B). Then

(j) jh  j j! i (2.25) f (s)  c (j + 1)(QAQBWAWB) ln (QAQBWAWB) + j 2 QAQBWAWB

for |s − 1|  1/ ln(qQAQBWAWB) and uniformly on j, where c is a suitable positive constant. Remark 2.2.9. We observe that in case (iii) we do not assume that χ is primitive. Q Remark 2.2.10. Condition i|αi(p)| ≤ 1 in case (iv) of the previous theorem can be Q  relaxed to i|αi(p)|  p . We remark that such condition appears to be essential in our proof, so that (ii) and (iii) are not particular cases of (iv).

Proof. (i) Let S be a finite set of primes independent of QA and WA that we will choose later on. We consider the quantity

X |an| S (x) := . S n n≤x (n,S)=1

The first step is proving that there exists a positive constant c1 such that

c1  (2.26) SS(x)  (QAWA) x for every  > 0. 36 2. THE SIEGEL ZERO

Proof. By Lemma 2.2.2 we know that ∞ X An L(s, A ⊗ A¯) =: ns n=1 has a polynomial growth on vertical lines, uniformly for σ in a bounded interval, i.e., that the estimate

mA⊗A¯ ¯ c1 (2.27) (s − 1) L(s, A ⊗ A) a,b (QAWA(|t| + 1))

holds when a < σ < b, with some c1 > 0. By Proposition 1.5.3 we have that |an| ≤ 2 1 + |an| ≤ 1 + An, so

X 1 + An XAn n r (2.28) S (x) ≤ ≤ log x + 1 −
2r = S n n 2x n≤x n≤2x 2rr! Z +i∞ L(s + 1, A ⊗ A¯)(2x)s log x + ds 2πi −i∞ s(s + 1) ··· (s + r) where r is a large parameter assuring the convergence of the integral. Hence, (2.26) follows immediately by (2.27).  The second step is suggested by Iwaniec’s work, i.e., showing that (2.29) S2(x) ≤ c xS (x2) for some c > 0 . S 2 S 2 Proof. By Proposition 1.4.5 we have

∗ ∗ |a nm | 2 X |anam| X 1 X X X R(d) d S (x) = ≤ R(d)|a nm | = S nm mn d d nm n,m≤x n,m≤x d|(n,m)dA n,m≤x d|(n,m)dA d (nm,S)=1 (nm,S)=1 d|nm (nm,S)=1 d|nm

 X∗ R(d) X |aA| ≤ max #{(n, m): n, m < x, nm = AD} . d A A,D≤x2 d≤x2 A≤x2 (d,S)=1 (A,S)=1 Therefore  X∗ R(d) (2.30) S2(x)  ln x S (x2). S d S d≤x2 (d,S)=1 Now we estimate the sum in (2.30). Recalling that R is a completely multiplicative function and that P∗ denotes a sum over the square-full divisors only, we get ∞ X∗ R(d) Y  R2(p) R3(p)  Y  R2(p) XR(p)j ≤ 1 + + + ··· ≤ 1 + . d p2 p3 p2 p d≤x2 p≤x2 p≤x2 j=0 (d,S)=1 p6∈S p6∈S

By Axiom (A4) and the definition of R(p), it follows that there exists c3 depending on dA 1/2 but independent of QA, WA such that R(p) ≤ c3p for every prime. Hence, the series 2.2. THE AXIOMATIC L∗-CLASSES 37

in the infinite product is always convergent if a convenient set S is chosen, for example 2 S = {p : p ≤ c3 + 1}. With this choice of S the estimate becomes 2 X∗ R(d) Y c5R (p) Y c6  1 +
≤ 1 +
 lnc6 x , d p2 p d≤x2 p≤x2 p≤x2 (d,S)=1 p6∈S and (2.29) is proved in the stronger form S2(x)  lnc6+1x S (x2). S S  By iterating (2.29) and using (2.26) we obtain

M M M M S2 (x) ≤ c x2 S (x2 ) ≤ c (Q W )c1 x2  for any M > 0, S M S A A M so, taking the 2 -th root with a suitable M = M(c1, ), we have  (2.31) SS(x)  (QAWAx) . Since the local factors are convergent for σ > θ by Axiom (A4) (see Remark 2.2.2), P∞ h the series h=0|aph |/p converges for any p and is bounded by a constant independent of QA and WA. Hence from (2.31) we get ∞ X |an| X |an| Y X |aph | (2.32) lnjn ≤ lnjx  lnjx S (x)  (Q W x) lnj x , n n ph S  A A n≤x n≤x p∈S h=0 (n,S)=1 uniformly on j. Deducing (2.22) from this estimate is now a standard matter. Let I(j)(x) be defined by 1 Z 2+i∞ f (j)(s + 1)xs I(j)(x) := ds 2πi 2−i∞ s(s + 1) ··· (s + r) with a large r to assure the convergence and independent of j by Remark 2.2.7. We remark that dj sm P (s, log n)
= j,m dsj ns ns j+1 m j with Pj,m(x, y) a polynomial bounded by m (j + 1)x y uniformly on m and j. More- over,

Z 2+i∞ m ( 1 s x
s 0 if n > x ds = rm n
r rm2r 2πi 2−i∞ s(s + 1) ··· (s + r) n  r! 1 + x ≤ r! unif. on r if n ≤ x, hence

mA r r 2 X |an| I(j)(x)  mj+1(j + 1) logj n  (j + 1)cj(Q W x) logj x A r! n  A A n≤x uniformly on j, with c > 0. We move the integration line to σ = −1/2. By the residue theorem and Remark 2.2.7 we obtain f (j)(1) j!(Q W )c1  I(j)(x) = + O A√ A . r! 2j x 38 2. THE SIEGEL ZERO

2c1+2 Therefore, for x = (QAWA) we get the estimate

(j) j  j j! f (1)  (j + 1)c (QAWA) ln (QAWA) + j uniformly on j, 2 QAWA with some c > 0. (j) P∞ f (j+u)(1) u At last, the power series expansion f (s) = u=0 u! (s − 1) gives (2.22) in the range |s − 1|  1/ ln(QAWA).

(ii) The proof is very similar to that of (i). We define

X An S (x) := S n n≤x (n,S)=1 with S a finite set of primes independent of QA and WA to be chosen later on. As before the first step is proving that there exists a positive constant c1 such that

c1  (2.33) SS(x)  (QAWA) x for any  > 0 . This can be done easily since inequalities (2.28) still hold. Moreover, we have assumed Axiom (A4) for L(s, A ⊗ A¯), so that again we can perform the second step, i.e., showing that (2.34) S2(x) ≤ c xS (x2) for some c = c (d ) > 0 . S 2 S 2 2 A Now, iterating (2.34) and using (2.33) we obtain

 (2.35) SS(x)  (QAWAx) . Since the local factors are convergent for σ > θ by Axiom (A4) (see Remark 2.2.2), P∞ h the series h=0 Aph /p converges for any p and is bounded by a constant independent of QA and WA. Hence from (2.35) we get ∞ X An X An Y X |aph | (2.36) lnjn ≤ lnj x  lnj x S (x)  (Q W x) lnj x , n n ph S  A A n≤x n≤x p∈S h=0 (n,S)=1 uniformly on j. The claim follows from this estimate as in the previous case.

(iii) We remark that by the complete multiplicativity of χ we have

∞ X Anχ(n) L (s, A ⊗ A¯) = , χ ns n=1 ¯ where An are the Dirichlet coefficients of L(s, A ⊗ A). As in case (i) we define

Z 2+i∞ (j) s (j) 1 fχ (s + 1)x Iχ (x) := ds . 2πi 2−i∞ s(s + 1) ··· (s + r) 2.2. THE AXIOMATIC L∗-CLASSES 39

Then

m ¯ r r A⊗Aχ 2 X |Anχ(n)| (2.37) I(j)(x)  mj+1 (j + 1) logj n ≤ χ A⊗A¯χ r! n n≤x m ¯ r r A⊗Aχ 2 X An mj+1 (j + 1) logj n  (j + 1)cj(Q W x) logj x , A⊗A¯χ r! n  A A n≤x uniformly on j, by (2.36) and |χ(n)| ≤ 1. If χ is primitive, we move the integration line to σ = −1/2. By the residue theorem and Remark 2.2.7 we obtain (j) f (1) j!(qQ W )c1  (2.38) I(j)(x) = χ + O A√ A . χ r! 2j x Nevertheless, if χ is induced by χ0 ( mod q0), we have 0 Y Y αi(p)¯αj(p)χ (p) L (s, A ⊗ A¯) = 1 − L 0 (s, A ⊗ A¯) =: G(s)L 0 (s, A ⊗ A¯). χ ps χ χ p|q/q0 i,j It is immediate verifying that |G(s)| < qc for some c > 0 in the strip 1/2 ≤ σ ≤ 3, so that (2.38) holds for general χ as well. 2c1+2 Taking x = (qQAWA) , our claim follows by (2.37) and (2.38).

(iv) Let A(n) and B(n) the totally multiplicative functions defined by

d d YA YB A(p) := αi(p), B(p) := βj(p). i=1 j=1 By Littlewood’s lemma quoted in Subsection 1.5.3 we have

2 ∞ 2 Y |A(p)|  X An Y |A(p)|  (2.39) 1 − = 1 − L(s, A ⊗ A¯) = pdAs ns pdAs p n=1 p ∞ 2 X |a(n1, . . . , nd −1)| A , 2 dA−1 s n ,··· ,n =1 (n1n2 ··· nd −1) 1 dA−1 A

2 ∞ 2 Y |B(p)|  X Bn Y |B(p)|  (2.40) 1 − = 1 − L(s, B ⊗ B¯) = pdBs ns pdBs p n=1 p ∞ 2 X |b(n1, . . . , nd −1)| B , 2 dB−1 s n ,··· ,n =1 (n1n2 ··· nd −1) 1 dB−1 B and ∞ Y A(p)B(p) X a(n1, . . . , nd−1)b(n1, . . . , nd−1) (2.41) 1 − L(s, A ⊗ B) = ds 2 d−1 s p (n1n ··· n ) p n1,··· ,nd−1=1 2 d−1 if dA = dB = d, 40 2. THE SIEGEL ZERO

∞ X a(n1, . . . , nd −1)b(n1, . . . , nd , 1, ··· , 1)A(nd ) (2.42) L(s, A ⊗ B) = A A A 2 dA s n ,··· ,n =1 (n1n2 ··· nd ) 1 dA A

if dA < dB, where the coefficients a(·) and b(·) are multiplicative in every entry. We assumed that L(s, A ⊗ A¯) and L(s, B ⊗ B¯) satisfy Axiom (A4), hence by (2.36), (2.39), (2.40) and the fact by the assumptions we have |A(n)|, |B(n)| ≤ 1 for every n, for dA ≤ dB we have 2 P Ad X |a(n1, . . . , ndA−1)| X d|n X An  (2.43) ≤  log x  (QAWAx) n n2 ··· ndA−1 n n 2 dA−1 1 2 dA−1 n≤x n≤x n1n ···n ≤x 2 dA−1 and

2 2 X |b(n1, . . . , nd , 1,..., 1)| X |b(n1, . . . , nd −1)| (2.44) A ≤ B ≤ n n2 ··· ndA n n2 ··· ndB−1 2 dA 1 2 dA 2 dB−1 1 2 dB−1 n1n ···n ≤x n1n ···n ≤x 2 dA 2 dB−1 P X d|n Bd X Bn  log x  (Q W x) . n n  B B n≤x n≤x

Assume now dA = dB = d. By (2.41) and (2.43) it follows that

X j 2 d−1
a(n1, . . . , nd−1)b(n1, . . . , nd−1) (2.45) ln n1n ··· n ≤ 2 d−1 n n2 ··· nd−1 2 d−1 1 2 d−1 n1n2···nd−1≤x 2 1/2 2 1/2  X |a(n1, . . . , nd−1)|   X |b(n1, . . . , nd−1)|  ≤ lnj x  n n2 ··· nd−1 n n2 ··· nd−1 2 d−1 1 2 d−1 2 d−1 1 2 d−1 n1n2···nd−1≤x n1n2···nd−1≤x  j  (QAQBWAWBx) ln x uniformly on j. We define Y A(p)B(p) 1 − L(s, A ⊗ B) =: L∗(s, A ⊗ B). pds p

1 By Remark 2.2.7 L(s, A ⊗ B) has a polynomial behaviour on the strip 1 − 2d < σ < 2, and the same holds for Q1 − A(p)B(p)/pds
since |A(p)|, |B(p)| ≤ 1 and d > 1 by hypothesis. Therefore, we have that the same holds for L∗(s, A ⊗ B). Hence we write

2+i∞ m ∗
(j) s 1 Z s A⊗B L (1 + s, A ⊗ B) x I(j)(x) := ds 2πi 2−i∞ s(s + 1) ··· (s + r) with a large r to assure the convergence, not depending on j. As in case (i), from (2.40) and (2.45) we get (j) j
 j (2.46) I (x)  (j + 1)c QAQBWAWBx ln x 2.2. THE AXIOMATIC L∗-CLASSES 41

uniformly on j. Moving the integration line to σ = −1/2d we have

m ∗
(j) s A⊗B L (1 + s, A ⊗ B) | j!(Q Q W W )c1  (2.47) I(j)(x) = s=0 + O A B A B , r! 2jx1/2d

2d(c1+1) for some positive constant c1. Choosing x = (QAQBWAWB) , our claim follows by (2.46), (2.47) and some easy algebraic manipulations.

Assume now dA < dB. Then, by (2.42) and (2.44), we get a(n , . . . , n )b(n , . . . , n , 1, ··· , 1)A(n ) X j 2 dA
1 dA−1 1 dA dA ln n1n2 ··· n ≤ dA n n2 ··· ndA 2 dA 1 2 dA n1n ···n ≤x 2 dA 2 1/2 2 1/2  X |a(n1, . . . , nd −1)|   X |b(n1, . . . , nd , 1, ··· , 1)|  ≤ lnj x A A  n n2 ··· ndA n n2 ··· ndA 2 dA 1 2 dA 2 dA 1 2 dA n1n ···n ≤x n1n ···n ≤x 2 dA 2 dA  j  (QAQBWAWBx) ln x

uniformly on j. The claim of case (iv) for dA < dB follows from this estimate.  Remark 2.2.11. Suppose that the coefficients of L(s, A) satisfy a stronger estimate than that one of Axiom (A4), i.e., that

1/j 1/4 (2.48) |sj(p)|  p for every 2 ≤ j ≤ dA. Then, consider 2 X |an| S (x) := . S n n≤x (n,S)=1 2 By Proposition 1.5.3 we have |an| ≤ An, so that
c  SS(x)  QAWA x holds from (2.27). Iwaniec’s trick is again applicable, and we obtain, similarly to (2.30), that  X∗ R2(d) S2(x)  ln3 x S (x2), S d S d≤x2 (d,S)=1 where the series is convergent by (2.48). From this we get

2 X |an| (2.49)  (Q W x) . n  A A n≤x Therefore, if estimate (2.48) holds for two L-functions, say L(s, A) and L(s, B), then (2.43) and (2.44) easily follow from (2.49), and upper-bounds (2.25) follows without as- suming Axiom (A4) for L(s, A ⊗ A¯) or L(s, B ⊗ B¯). This remark will be important in Subsection 2.3.2 below. 42 2. THE SIEGEL ZERO

2.2.4. A general Hecke-type theorem. The results in this subsection give a rela- tion between the existence of the Siegel zero and lower bounds for L(1, A). They extend to the general setting of L∗-classes the classical result of Hecke about Dirichlet L-series (see the introduction to this chapter). Proposition 2.2.2. Let L(s, A) be in an L∗-class.

(i) If L(s, A) is positive and mA = 1, then  Ress=1 L(s, A)  δA(1 − βA) + (1 − δA)(QAWA) .

(ii) If L(s, A) is regular (mA = 0) and δA = 1 −c L(1 + iγA, A)  |1 − βA|(QAWA) . Moreover, if L(s, A ⊗ A¯) satisfies (A4) then − L(1 + iγA, A)  |1 − βA|(QAWA) .

(iii) If L(s, A) is regular (mA = 0) and δA = 0 −c L(1 + it, A)  (QAWA) . Moreover, if L(s, A ⊗ A¯) satisfies (A4) then − L(1 + it, A)  (QAWA) . Here, c is a suitable positive constants,  is an arbitrary positive constant, t  1/ log(QAWA), and ρA = βA + iγA. We remark that even if we assume that the Siegel zero does not exist, we are unable to obtain logarithmic lower-bounds for L(s, A), contrary to the case where the Ramanujan hypothesis is assumed. This limit is clearly due to Theorem 2.2.1. We give a sketch of the proof since it is a simplified version of that in [7]. Proof. The proof of this result follows that of Theorem 4 of [7], hence we present only the most interesting case, i.e., case (ii). We define ∞ X bn F (s) := ζ(s)L(s + iγ , A)L(s − iγ , A¯)L(s, A ⊗ A¯) =: . A A ns n=1

As it is easy to verify, b1 = 1 and bn ≥ 0 for every n. By Perron’s formula we have ∞ 1 X 1 Z 2+i∞ ≤ b (t)n−βA e−n/x = F (β + s)Γ(s)xsds = 2 n 2πi A n=1 2−i∞ Z −1/2+i∞ 1 s R(x) + F (βA + s)Γ(s)x ds 2πi −1/2−i∞ s where R(x) is the residue of F (βA + s)Γ(s)x at the double pole s = 1 − βA; here we have used F (βA) = 0 to cancel the simple pole of Γ(s) at s = 0. By Lemma 2.2.2 we get 1 (Q W )c1  (2.50) ≤ R(x) + O A√ A with c > 0. 2 x 1 2.2. THE AXIOMATIC L∗-CLASSES 43

2c1+1 We take x = (QAWA) in (2.50), so that

2c1+1 (2.51) 1  R((QAWA) ).

Moreover, x1−βA  1 by definition of Siegel zero. A direct computation gives

¯ 2 1−βA R(x) =r−1(ζ)r−1(A ⊗ A)[ |L(1 + iγA, A)| Γ(1 − βA)x log x+

2 0 1−βA |L(1 + iγA, A)| Γ (1 − βA)x +

0 ¯ 1−βA L(1 + iγA, A)L (1 − iγA, A)Γ(1 − βA)x +

¯ 0 1−βA L(1 − iγA, A)L (1 + iγA, A)Γ(1 − βA)x ]+

¯ 2 1−βA r−1(ζ)r0(A ⊗ A) |L(1 + iγA, A)| Γ(1 − βA)x +

¯ 2 1−βA r0(ζ)r−1(A ⊗ A) |L(1 + iγA, A)| Γ(1 − βA)x , ¯ ¯ where rj(ζ) and rj(A ⊗ A) are the Laurent coefficients at s = 1 of ζ(s) and L(s, A ⊗ A), respectively. By Theorem 2.2.1 we get

2c1+1 ¯ ¯
 R((QAWA) )  |r−1(A ⊗ A)| + |r0(A ⊗ A)| (QAWA) · h L(1 + iγ , A) L(1 + iγ , A) 2i A + A , 1 − βA 1 − βA and by Remark 2.2.7 we have

h L(1 + iγ , A) L(1 + iγ , A) 2i 2c1+1 A A c2 (2.52) R((QAWA) )  + (QAWA) 1 − βA 1 − βA for some c2 > 0. From (2.51) and (2.52) we obtain

L(1 + iγ , A) L(1 + iγ , A) 2 −c2 A A (QAWA)  + . 1 − βA 1 − βA At least one term on the right-side is  than the left-side, so that the first claim follows with c = c2 or c = c2/2. ¯ If L(s, A⊗A) verifies Axiom (A4), then we can choose c2 =  in (2.52) by Theorem 2.2.1 again, so that the second claim of case (ii) follows as well. 

We remark that in the above proof, where the best value for the constant c is not 0 required, we do not need any estimate for L /L(1 + iγA, A). From Proposition 2.2.2 and Theorem 2.2.1 we easily get the following

Corollary 2.2.1. Let L(s, A) be regular (mA = 0), δA = 1 and assume that L(s, A⊗ A¯) satisfies Axiom (A4). Then

− − L(1 + iγA, A)  (QAWA) ⇐⇒ |1 − βA|  (QAWA) . 44 2. THE SIEGEL ZERO

2.3. Siegel-type theorems In the previous sections we developed the analytic tools necessary to prove a general Siegel-type theorem we state now. ∗ Theorem 2.3.1. Let Md a subset of an L -class, such that

(i) every function of Md is entire, totally real and of analytical degree d, (ii) for every choice of L(s, A), L(s, B) ∈ Md, there exists a function L(s, C) in the ∗ same L -class with analytic degree dC and order of pole mC uniformly bounded over Md, such that -) the following function is positive F (s) := LmC (s, A)L(s, B)L(s, C) , -) the following estimates hold

(2.53) log QC  log QA + log QB , log WC  log WA + log WB . Then we have
− (2.54) L(1, A)  QAWA for every  > 0. The proof of this theorem follows that one of the original Siegel’s theorem we have reproduced in Section 2.1.

Proof. Let 0 <  < 1/2. We can assume that there exists a function L(s, A) ∈ Md with a zero β ∈ (1 − , 1), otherwise (2.54) is a consequence of Corollary 2.2.1. Hence, we choose B = A in the definition of F (s), so that F (β) = 0, and we write ∞ X an F (s) = ns n=1

with an ≥ 0, a0 = 1 by hypothesis. Then, for x > 1 we have ∞ X 1 Z 2+i∞ (2.55) e−1/x ≤ a n−β e−n/x = F (s + β )Γ(s)xsds = n 2πi  n=1 2−i∞ Z −1+i∞ s
1 s Ress=1−β F (s + β)Γ(s)x + F (s + β)Γ(s)x ds 2πi −1−i∞ by the residue theorem and Lemma 2.2.2; here the simple pole of Γ(s) at s = 0 is canceled by the zero of F (s) at s = β. We remark that the constant appearing in Lemma 2.2.2 depends only of the interval and of the analytic degree of the function involved. In our present case, β ∈ (1/2, 1) so it mC is uniformly bounded on , and the analytic degrees of L (s, A), L(s, A) and L(s, C) are independent of  by hypothesis. Hence, by Lemma 2.2.2 and assumptions (2.53), there exists a positive and independent of  constant c such that
c c F (s + β)  QAQA WAWA (1 + |t|) on the vertical line s = −1 + it, so that (2.55) gives (Q Q W W )c  1  Res F (s + β )Γ(s)xs
+ O A A A A , s=1−β   x 2.3. SIEGEL-TYPE THEOREMS 45

c+1 and for x = QAQA WAWA we get s
(2.56) 1  Ress=1−β F (s + β)Γ(s)x . Moreover, since by hypothesis L(s, A) divides F (s) with multiplicity at least equal to the order of pole at s = 1 of F (s), a direct computation shows that

s
1−β Ress=1−β F (s + β)Γ(s)x = L(1, A)x P(, A, A, C, log x) where P(, A, A, C, log x) is a polynomial of log x and of the Laurent coefficients at s = 1 of L(s, A), L(s, A) and L(s, C). By Theorem 2.2.1, hypothesis (2.53) and the choice of x, we have   P(, A, A, C, log x)  QAWA ,

1−β
(c+1) x  QAWA , so that s

 (2.57) Ress=1−β F (s + β)Γ(s)x  L(1, A) QAWA . Theorem follows from (2.56) and (2.57).  As it is clear by Theorem 2.3.1, a Siegel-type theorem can be deduced if a suitable positive convolutions is found. The next subsections deal with the problem of building positive convolutions in certain L∗-classes, so that Siegel-type theorems will follow for such L∗-classes. We recall that the generalized von Mangoldt function Λn(A) is defined by 0 ∞ L X Λn(A) − (s, A) = , L ns n=1 and that L(s, A), or A, is positive when Λn(A) ≥ 0 for every n. By considering the Euler product of L(s, A + B) and L(s, A ⊗ B), it is almost immediate verifying that m Λn(A) = 0 if n 6= p , ¯ Λn(A) = Λn(A) ,

Λn(A + B) = Λn(A) + Λn(B) ,

Λpm (A ⊗ B) log p = Λpm (A)Λpm (B). These identities immediately give the following positive convolutions.

(i) If χ is a real Dirichlet character, then Λn(1 + χ) ≥ 0 for every n. In general, if L(s, A) is a totally real function satisfying the Ramanujan hypothesis, then Λn(dA + A) ≥ 0 for every n. ¯ (ii) Λn(A ⊗ A) ≥ 0 for every n: a fact we have already used. (iii) If A and B are positive, then Λn(A + B) and Λn(A ⊗ B) are positive too. Cases (i) and (ii) provide the ”basic” positive convolutions, while case (iii) provides two operations to obtain other positive convolutions. For special L∗-classes having an algebraic structure, there are special positive convolutions which escape from the above three cases (see for example Subsection 2.3.4), but in our general setting it appears to be difficult finding other basic positive convolutions. 46 2. THE SIEGEL ZERO

Definition 2.3.1. We say that an L∗-class is orthogonal if for every irreducible A and B we have ( 1 when A = B¯, m = m = A⊗B A⊗B] 0 when A 6= B¯.

Remark 2.3.1. The equality Aχ = A ⊗ χ has interesting consequences in orthogonal and twist closed L∗-classes. In fact, let L(s, A) and L(s, B) be irreducible functions of such a class. If A 6= 1, by orthogonality it follows that L(s, A) is entire. Moreover, by orthogonality again, it can exist only a primitive character χ at most such that Lχ(s, A) has a pole at s = 1, and the same holds for Lχ(s, A ⊗ B). We remark that the orthogonality, as defined here, is a different form of Selberg’s orthogonality conjecture (see Selberg [56]). In this section we consider only orthogonal L∗-classes, since this property is essential for our approach to Siegel-type theorems.

2.3.1. Siegel for twists by Dirichlet characters. Assume that the L∗-class is twist closed and orthogonal. Let L(s, A) be a fixed irreducible and totally real function and let χ be a primitive real character. From Proposition 2.2.1 we know that the Siegel zero βχ of Lχ(s, A), if it exists, is real. Hence, let χ and χ0 be distinct, real, primitive and non-trivial characters with modu- ∗ lus sufficiently large to assure that Aχ and Aχ0 belong to the same L -class (see Defini- tion 2.2.4) and be entire (see Remark 2.3.1). Consider F (s) :=Ls, (1 + χ) ⊗ (1 + χ0) ⊗ (1 + A) ⊗ (1 + A)
= ζ(s)L(s, χ)L(s, χ0)L(s, χχ0)· 2 2 2 2 L (s, A)Lχ(s, A)Lχ0 (s, A)Lχχ0 (s, A)· L(s, A ⊗ A)L(s, A ⊗ Aχ)L(s, A ⊗ Aχ0)L(s, A ⊗ Aχχ0). Since 1 + χ, 1 + χ0 and (1 + A) ⊗ (1 + A) are positive convolutions, F (s) is positive as well. We observe that Lχ(s, A⊗A) is regular by the orthogonal property since by hypothesis 0 A 6= Aχ; similarly for Lχ0 (s, A ⊗ A) and Lχχ0 (s, A ⊗ A). Also, L(s, χ), L(s, χ ) and L(s, χχ0) are regular for the same reason. Therefore, only ζ(s), L(s, A) and L(s, A ⊗ A) contribute to the order of pole at s = 1 of F (s). If A = 1, F (s) has a pole of order four at most and it is divided by Lχ(s, A) = L(s, χ) with multeplicity four at least. If A= 6 1, L(s, A) is entire by Remark 2.3.1 so that F (s) has a pole of order two at most and is divided by Lχ(s, A) with multeplicity two. In both cases Theorem 2.3.1 proves the following Siegel-type theorem for twists of a fixed function L(s, A). Theorem 2.3.2. Let L(s, A) be an irreducible totally real L-function in a twist closed orthogonal L∗-class. Assume that L(s, A⊗A) satisfies Axiom (A4) and let χ be a primitive real character with a sufficiently large modulo q so that Lχ(s, A) is regular. Then − Lχ(1, A) A, q for every  > 0. 2.3. SIEGEL-TYPE THEOREMS 47

We observe that if L(s, A ⊗ A) satisfies Axiom (A4), then Lχ(s, A ⊗ A) satisfies (A4) too, since χ is completely multiplicative; hence cases (i) and (iii) of Theorem 2.2.1 assure the required upper-bounds for Lχ(1, A) and Lχ(1, A ⊗ A). Now we deal with the case of L(s, A ⊗ B) in a similar way, where B is irreducible and totally real. Consider

F (s) :=Ls, (1 + χ) ⊗ (1 + χ0) ⊗ (A + B) ⊗ (A + B)
= L(s, A ⊗ A)L2(s, A ⊗ B)L(s, B ⊗ B)· L(s, A ⊗ Aχ)L2(s, A ⊗ Bχ)L(s, B ⊗ Bχ)· L(s, A ⊗ Aχ0)L2(s, A ⊗ Bχ0)L(s, B ⊗ Bχ0)· L(s, A ⊗ Aχχ0)L2(s, A ⊗ Bχχ0)L(s, B ⊗ Bχχ0).

If A= 6 B, F (s) has pole at s = 1 of order two at most and it is divided by Lχ(s, A ⊗ B) with multiplicity 2; if A = B, F (s) has pole of order four at most and is divided by Lχ(s, A ⊗ A) with multiplicity four. This positive convolution and Theorem 2.3.1 prove the following Siegel-type theorem for twists of a fixed function L(s, A ⊗ B).

Theorem 2.3.3. Let A and B be totally real irreducible elements of an orthogonal ∗ QdA QdB and twist closed L -class, such that i=1|αi(p)|, j=1|βj(p)| ≤ 1. Moreover, assume that L(s, A ⊗ A) and L(s, B ⊗ B) satisfy Axiom (A4) and let χ be a primitive real character with a sufficiently large modulo q so that Lχ(s, A ⊗ B) is regular. Then

− (2.58) Lχ(1, A ⊗ B) A,B, q for every  > 0. The additional conditions of this theorem are necessary to apply case (iv) of Theo- rem 2.2.1 to get the required upper bound of (s − 1)mA⊗B L(1, A ⊗ B).

Example 2.3.1. Consider the L∗-class of Dirichlet L-functions associated with primi- tive Dirichlet characters. It is well known that this is an orthogonal L∗-class. If we choose A = 1 in Theorem 2.3.2, we obtain the original Siegel theorem.

Example 2.3.2. Let Sk,new be the set of holomorphic cuspidal newforms of weight k for a congruence group Γ0(N) and with some multiplicative system (see for example Miyake [40]). Consider the set of normalized L-functions associated with such cusp forms. Then, the classical theory assures that this is an L∗-class. In particular, these functions satisfy the Ramanujan hypothesis by the work of Deligne [11], so that Axiom (A4) is trivially satisfied. Moreover, Axiom (A5) and the orthogonal property are provided by the well known work of Rankin and Selberg. Further, the class is twist closed (Theorem 4.3.2 of [40]). For these functions dA = dA = 2, QA is essentially the level N and WA the weight k, so that, if we choose as L(s, A) the L-function associated with an eigenfunction f for the full modular group SL2(Z) with trivial multiplicative system, from Theorems 2.3.2 and 2.3.3 we get the following corollary. 48 2. THE SIEGEL ZERO

Corollary 2.3.1. Let f and g be chosen as above, with possibly f = g, and χ be a real primitive character modulo q. Then

− (2.59) Lχ(1, f) f, q − (2.60) Lχ(1, f ⊗ g) f,g, q for every  > 0. Estimates (2.59) and (2.60) for f = g have been already proved by Golubeva and Fomenko [21] (Theorems 5 and 6). The lower-bound (2.60) for f 6= g has been proved by Ichihara [26], by a technical variant of the above method.

We remark that Deligne’s result is not necessary here. In fact, let α1(p) and α2(p) be the local coefficients at a generic prime p of a newform as above. Then α1(p)α2(p) = 1, 1/2 and an easy and well known computation shows that |αj(p)| ≤ p (see Miyake [40], Corollary 2.1.6). Moreover, df = 2, hence only the second symmetric polynomial needs to be considered for L(s, A). But s2(A) = 1, so that we immediately see that L(s, A) 2 satisfies Axiom (A4). Moreover, L(s, A ⊗ A) has degree 4, with local coefficients α1, 1, 1, 2 α2. Hence, 2 2 s2(A ⊗ A) = 2[α1 + 1 + α2], 2 2 s3(A ⊗ A) = α1 + 2 + α2 ,

s4(A ⊗ A) = 1 ,

1/2 therefore estimate |αj(p)| ≤ p is sufficient to assure Axiom (A4) for L(s, A ⊗ A) as well. Example 2.3.3. Let f be a Maass form, i.e., a cusp forms of weight zero for a fixed congruence group Γ0(N) with trivial multiplicative system and eigenfunction with eigen- value λ of the hyperbolic laplacian. It is well known that f can be viewed as an unitary cuspidal automorphic representation of GL(2) (see Gelbart [15]) and that we can asso- ciate to f a totally real L-function L(s, f). Moreover, L(s, f) has an Euler product of degree 2, holomorphic continuation to C and satisfies a functional equation of the form of Axiom (A3), with the main parameter Q depending on the level N and the weight W depending on λ (see Remark 2.2.4). Further, L(s, f ⊗ f) exists and satisfies Axioms (A1)-(A3) by a general result of Moeglin and Waldspurger [41] about the Rankin-Selberg convolution of unitary cuspidal representations of GL(r). It is conjectured that L(s, f) satisfies the Ramanujan hypothesis, but the best estimate for the local coefficients of L(s, f) is

1/5 (2.61) α1(p)α2(p) = 1 , |α1(p)|, |α2(p)| < p , proved by Shahidi [59]. Hence, as in Example 2.3.2, equation α1(p)α2(p) = 1 assures Axiom (A4) for L(s, f) and (2.61) assures Axiom (A4) for L(s, f ⊗ f). Therefore, the analytic properties giving Corollary 2.3.1 are satisfied by these functions as well, so that the lower bounds (2.59) and (2.60) hold also in the case of f and g Maass forms. 2.3.2. Siegel for regular components. Let L(s, A) be an irreducible totally real function of an orthogonal L∗-class. Hence L(s, A ⊗ A) has a simple pole at s = 1. We 2.3. SIEGEL-TYPE THEOREMS 49

assume that ζ(s) factorizes L(s, A ⊗ A) in the class, i.e., that there exists a function ∗ L(s, RA) in the same L -class of A such that

L(s, A ⊗ A) = ζ(s)L(s, RA).

Definition 2.3.2. We say that RA is the regular component of A ⊗ A. Let A and B be totally real and irreducible functions of the L∗-class, with A 6= B and satisfying the above property. Define F (s) :=Ls, (A ⊗ A) ⊗ (B ⊗ B)
=
L s, (1 + RA) ⊗ (1 + RB) = ζ(s)L(s, RA)L(s, RB)L(s, RA ⊗ RB).

Then, F (s) is positive with a simple pole at s = 1, since A 6= B implies RA 6= RB and the ∗ L -class is orthogonal. Moreover, L(s, RA) divides F (s). Theorem 2.3.1 therefore proves the following Siegel-type theorem for L(s, RA). Theorem 2.3.4. In addition to the above hypotheses about A and the L∗-class, assume QdA that i=1|αi(p)| ≤ 1 for every prime p and that L(s, RA⊗RA) satisfies Axiom (A4). Then
− L(1, RA)  QRA WRA for every  > 0. As before, the additional conditions of this theorem are necessary to apply case (iv) of Theorem 2.2.1 to get the required upper-bound of L(1, RA ⊗ RB). By Remark 2.2.11, we can prove in a similar way the following variant of this theorem. Theorem 2.3.5. In addition to the above hypotheses about A and the L∗-class, assume QdA that i=1|αi(p)| ≤ 1 for every prime p and that 1/j 1/4 |sj(p)|  p for j = 2,..., dA. Then
− L(1, RA)  QRA WRA for every  > 0. Example 2.3.4. We pursue Example 2.3.3, observing that L(s, f ⊗ f) can be easily factorized as (2.62) L(s, f ⊗ f) =: ζ(s)L(s, sym2 f) σ > 1 , an identity that can also be assumed as definition of L(s, sym2 f). Gelbart and Jacquet [16] have proved the ”lift conjecture” for the morphism sym2 (see [16] for definitions and results), obtaining that (2.62) is actually a factorization in the set of unitary cuspidal au- tomorphic representations. This means that sym2 f is an irreducible cuspidal automorphic representation of GL(3), a fact that immediately gives Axioms (A1)-(A3) for L(s, sym2 f). It follows that sym2 f is the regular component of f. Moreover, L(s, sym2 f) has Euler 2 2 product of degree 3, with α1(p), 1, α2(p) as local coefficients, hence 2 2 2 s2(sym f)(p) = α1(p) + 1 + α2(p), 2 s3(sym f)(p) = 1 , so that (2.61) gives Axiom (A4) for L(s, sym2 f). In view of our previous discussion, we can form an L∗-class by the set of L(s, f) and L(s, sym2 f) functions. Such an L∗-class is orthogonal by [41]. 50 2. THE SIEGEL ZERO

It is almost immediate to verify that (2.61) is not sufficiently strong to assure Axiom (A4) for L(s, sym2 f ⊗sym2 g). Hence Theorem 2.3.4 is not applicable. Nevertheless, from (2.61) we immediately get 2 1/2 1/5 |s2(sym f)(p)|  p , 2 1/3 |s3(sym f)(p)|  1 , hence the hypotheses of Theorem 2.3.5 are satisfied, and we obtain the Siegel-type lower- bound 2 − (2.63) L(1, sym f)  (Nλ) . This estimate is the main result of Hoffstein and Lockhart [24]. 2.3.3. Siegel under Ramanujan hypothesis. Consider an orthogonal L∗-class such that the Ramanujan hypothesis holds for its elements. Moreover, assume that L(s, A ⊗ A) belongs to the same L∗-class of A for every totally real element A. This fact assures, by Axiom (A5), that L(s, A ⊗ A ⊗ B) satisfies Axioms (A1)-(A4), so that if d d YA YB |αi(p)| ≤ 1 , |βj(p)| ≤ 1 for every p, i=1 j=1 then (j)  mA⊗A⊗B

(s − 1) L(s, A ⊗ A ⊗ B) |s=1 j, QAWAQBWB for every  > 0, by case (iv) of Theorem 2.2.1. Let A and B be distinct, irreducible, regular and totally real elements of this L∗-class, with dA = dB = d > 1. We consider the positive convolution F (s) :=Ls, (1 + A) ⊗ (1 + A) ⊗ (d + B)
= ζ(s)L2(s, A)L(s, A ⊗ A))dL(s, B)L2(s, A ⊗ B)L(s, A ⊗ A ⊗ B).

The condition d > 1 gives B 6= A ⊗ A, so that mA⊗A⊗B = 0 by orthogonality, and F (s) has a pole at s = 1 of order 2d at most. Moreover, both L(s, A) and L(s, B) divide F (s), and L(s, A) divides F (s) with multiplicity 2d. Therefore, by Theorem 2.3.1 we can prove the following Siegel-type theorem Theorem 2.3.6. In addition to the above conditions about A and the L∗-class, assume QdA that i=1|αi(p)| ≤ 1 for every prime p. Then
− L(1, A)  QAWA for every  > 0. Example 2.3.5. As in Example 2.3.2, consider the L∗-class of L-functions associated with holomorphic cuspidal newforms of weight k for a congruence group Γ0(N) and with trivial multiplier system so that α1(p)α2(p) = 1. As we have already recalled, Ramanujan hypothesis holds by Deligne’s work [11]. Moreover, the holomorphic cusp functions can be considered as unitary cuspidal automorphic representations of GL(2) as well (see [15]). Hence, the factorization L(s, f ⊗ f) = ζ(s)L(s, sym2 f) holds, where sym2 f is an unitary cusp representation of GL(3) exactly as in Example 2.3.4. 2.3. SIEGEL-TYPE THEOREMS 51

It follows that L(s, f ⊗ f ⊗ g) = L(s, g)L(s, g ⊗ sym2 f), the hypotheses of Theorem 2.3.6 being satisfied by Moeglin and Waldspurger’s work [41]. We therefore get Corollary 2.3.2. Let f be an holomorphic cuspidal newform of weight k, level N and trivial multiplier system. Then − (2.64) L(1, f)  (Nk) for every  > 0. As far as the k-aspect is concerned, this estimate has been already proved by Golubeva and Fomenko [22]. Nevertheless, the fact that L(s, f⊗f⊗g) is regular and satisfies Axioms (A1)-(A3) as a consequence of [16] and [41] is apparently unknown to the authors, which state it among the hypotheses of their Theorem 2. 2.3.4. Further explicit results. Let f and g be holomorphic or Maass cusp forms as in Examples 2.3.3, 2.3.4 and 2.3.2. The results presented in these examples show that we can apply Theorem 2.3.3 with A = f and B = sym2 f, or A = f and B = sym2 g, or A = sym2 f and B = sym2 g, thus obtaining the following Siegel-type lower-bounds. Corollary 2.3.3. Let f and g be as above, f 6= g, and χ be a real primitive character modulo q. Then 2 − (2.65) Lχ(1, f ⊗ sym f) f, q , 2 − (2.66) Lχ(1, f ⊗ sym g) f,g, q , 2 2 − (2.67) Lχ(1, sym f ⊗ sym g) f,g, q . Remark 2.3.2. It is easy to see that L(s, f ⊗ sym2 f) = L(s, f)L(s, sym3 f) σ > 1 , where L(s sym3 f) is the function investigated by Shahidi [58]. Since the upper-bound  Lχ(1, f) f, q holds, from (2.65) we get 3 − (2.68) Lχ(1, sym f) f, q for every  > 0.

Now we turn to non-existence results for the Siegel zero. The non-existence of Siegel- zeros for not self-dual cuspidal automorphic representations π of GL(r) is easily proved, since such representations have complex L-functions and Proposition 2.2.1 excludes the existence of the Siegel zero in this case. As already recalled in Section 2.1, Goldfeld, Hoffstein and Lieman [20], Hoffstein and Ramakrishnan [25] and Banks [1] have recently proved that the Siegel-zero does not exist for general unitary cuspidal automorphic representations of GL(2) and GL(3). The technique of such non-existence proof is based on the following Lemma. Let L(s, A) be a positive L-function of an L∗-class. Then there exists an effective positive constant c such that L(s, A) has at most mA real zeros in the interval (1 − c , 1). log QAWA 52 2. THE SIEGEL ZERO

Similar lemmas are well known in literature (see the lemma of [20], for example). We state it here for reference, and we refer to [20] for its proof. The above Lemma proves the non-existence of Siegel-zeros for self-dual π (having therefore a totally real L-function) as soon as a positive convolution F (s) containing L(s, π) with multiplicity larger than its order of pole is found. In fact, the Siegel zero for L(s, π) would be a zero of F (s) of multiplicity larger than its order of pole, a contradiction by the above Lemma. As shown in [25], the existence of such a positive convolution is a consequence of the following conjecture, already quoted on page 22. Conjecture (Langlands [39]). For every cusp form π on GL(r) and π0 on GL(r0), 0 0 there exists an automorphic representation π  π of GL(rr ) whose L-function is the Rankin-Selberg convolution of L(s, π) and L(s, π0). In fact, this conjecture has three important consequences: ∗ Multiple Rankin-Selberg convolutions can be performed, by the result of Moeglin and Waldspurger [41] about the Rankin-Selberg convolution of two unitary au- tomorphic representations. 0 0 0 ∗ If r < r , L(s, π) divides L(s, π ) ⇐⇒ L(s, π  π ) has a pole at s = 1. ∗ Let π be self-dual. A component τ of π  π different from 1 and π can be always found. From these three facts it follows that L(s, π) divides L(s, π  τ), since L(s, π  (π  τ)) = L(s, (π  π)  τ) and L(s, (π  π)  τ) has a pole at s = 1 by definition of τ. Therefore, there exists an automorphic representation ρ such that L(s, π  τ) =: L(s, ρ)L(s, π) . Hence, in [25] the following positive convolution is considered. F (s) :=L(s, (1 + π + τ) ⊗ (1 + π +τ ¯)) = 2 ζ(s)L(s, π  π)L(s, τ  τ¯)L(s, τ)L(s, τ¯)L (s, π)L(s, π  τ)L(s, π  τ¯) = 4 ζ(s)L(s, π  π)L(s, τ  τ¯)L(s, τ)L(s, τ¯)L(s, ρ)L(s, ρ¯)L (s, π). F (s) has therefore a pole of order 3 at s = 1, by the poles of the diagonal terms ζ(s), L(s, π  π) and L(s, τ  τ¯). Moreover, every L-function appearing in F (s) is regular on C\{1} so that a Siegel-zero of L(s, π) would be a zero of F (s) with at least multiplicity 4: a contradiction by the above Lemma. For r = 2 the Langlands conjecture has been proved (see [16]), and for r = 3 good analytic properties and zero-free regions have been proved by Bump and Ginzburg [6] and Banks [1], so that the non-existence of the Siegel-zero for cuspidal automorphic represen- tations of GL(2) and GL(3) has been proved unconditionally with a similar approach.

In Examples 2.3.3, 2.3.4 and 2.3.2 we have already mentioned the fact that if f is a Maass or an holomorphic cusp forms, then f and sym2 f can be considered as unitary cuspidal automorphic representations of GL(2) and GL(3), respectively. Therefore, the 2.3. SIEGEL-TYPE THEOREMS 53

Siegel-zero does not exist for L(s, f) and L(s, sym2 f), so that estimates (2.59), (2.60) with f = g, (2.63) and (2.64) are now trivial by Proposition 2.2.2. Nevertheless, estimates (2.60) for f 6= g and (2.66)-(2.68) (involving functions that are unitary cuspidal automorphic irreducible representations of GL(4), GL(6), GL(9) and GL(4), respectively, if Langlands conjecture is correct) are non-trivial.

CHAPTER 3

Existence of a singularity for certain functions of degree 1

3.1. Introduction In this chapter we present a result on the structure of a different class of L-functions. The analytic axioms defining this class are similar but remarkably weaker than those of an L∗-class. Moreover, the closeness under the twist f ⊗ χ of an element f by every primitive character χ is assumed, with weak uniformity on χ of the analytic properties of such twists. At last, an Euler product representation of polynomial form is assumed. In this class there are elements f of arithmetic degree d = 2 and 3 such that f ⊗ χ is entire for every χ, and conjecturally this is also true for any d ≥ 2. Our result, Theorem 3.2.1 below, shows that this property does not hold for functions of degree 1, i.e., for every f of degree 1 there exists a primitive character χf such that f ⊗ χf has a pole at s = 1. This is compatible with the unproved conjecture that Dirichlet L-functions and their shifts are the only elements of degree 1 of this class. The proof of this theorem relies on an analytic lemma about the mean value of the coefficients of the convolution f ⊗ g, where g has degree d ≥ 2 and has regular twist g ⊗ χ for every χ, and on Proposition 1.2.1 giving an Ω estimate for the coefficients of the Dirichlet series of g. The analytic lemma is proved following Duke and Iwaniec [14] where the convolution of functions of degree 2 and 3 is investigated. The material presented here forms the body of paper [43].

3.2. Definitions and results

Given an integer d ≥ 1, we consider the class Cd of functions with the following properties:

∗ (arithmetical conditions) if f∈Cd, then

d Y Y −s −1 f(s)= (1 − αj(p)p ) p j=1

with |αj(p)| ≤ 1 for every j, p. Hence f has a Dirichlet series representation

∞ X −s f(s) = ann , n=1 absolutely convergent for σ > 1. 55 56 3. EXISTENCE OF A SINGULARITY FOR CERTAIN FUNCTIONS OF DEGREE 1

∗ (analytic conditions) for every integer q ≥ 1 and every primitive character χ mod q, the twisted function ∞ X −s (f ⊗ χ)(s) := χ(n)ann n=1 has continuation to C as a meromorphic function with a pole at s = 1 at most; moreover, (s − 1)m(f ⊗ χ)(s) is an entire function of finite order for some integer m, and f ⊗ χ satisfies a functional equation of type d(s−1/2) f ¯ (f ⊗ χ)(1 − s) = q Φχ(s)(f ⊗ χ¯)(s) ¯ P −s f where f(s) := n a¯nn ,Φχ(s) is an holomorphic function in σ > 0 and satisfies the estimate f B(σ,χ) |Φχ(s)| < c(σ, χ)|t| for |t| ≥ 1 on each vertical line σ + it, for some constants c(σ, χ), B(σ, χ) > 0. Moreover, we assume that there existsσ ˜ > 0 such that c(σ, χ) = c(σ) and B(σ, χ) = B(σ) for σ > σ˜. f σ ∗ In addition, for f ∈ C1 we assume that Φχ(s)  |t| uniformly for |t| > 1 and σ sufficiently large.

Remark 3.2.1. The above conditions are inspired by the work of Duke and Iwaniec [14].

0 Remark 3.2.2. With these hypotheses, Cd0 ⊆ Cd when d ≤ d, so the really interesting 0 parameter associated with f ∈ Cd is d(f) := min{d : f ∈ Cd0 }. In the following we will assume that d(f) = d when we will write f ∈ Cd. Remark 3.2.3. The third condition is compatible with the previous two and is nec- essary in a technical point of Subsection 3.3.2. S Remark 3.2.4. The set d Cd has algebraic structure, provided by the product and the Rankin-Selberg convolution: in fact let f ∈ Cd and g ∈ Cd0 , then the identity (fg)⊗χ = (f ⊗χ)(g ⊗χ) shows that fg ∈ Cd+d0 . Moreover, assuming that f ⊗g satisfies the analytic conditions, then f ⊗ g ∈ Cdd0 . It is not completely trivial to show that the usual Dirichlet L-functions L(s, κ) belong to C1, the non-trivial part being the existence of the χ-uniform estimate for f ⊗ χ = L(s, κχ); we prove this in Appendix. Likewise, it can be proved that the normalized L-functions associated with holomorphic newforms for the Hecke group Γ0(N) with mul- tiplier κ are in C2: in this case we know that the twisted function L ⊗ χ is again a ˜ normalized L-function associated with a newform for a Γ0(N) and a new multiplier, so in this case f ⊗ χ is always an entire function (see Miyake [40], Theorem 4.3.12). Moreover, let L(s) be a normalized function associated with an holomorphic newform m for SL2(Z) and let L(s, sym ) be the m-symmetric function generated by L(s), introduced by Serre in connection with the Sato-Tate conjecture. For m ≥ 1 the Langlands program m m implies that L(s, sym ) ∈ Cm+1 and that the twist L(s, sym ) ⊗ χ is entire for all χ. 3.2. DEFINITIONS AND RESULTS 57

For small values of m these conjectures are consequences of important results proved in literature. In particular, they are true for m = 1 (case already quoted) and for m = 2 (from Shimura [60]). They are ’almost’ true for m = 3, 4, 5 too, in the sense that for such values of m the functional equation and the meromorphic continuation to C have been established (Shahidi [57, 58]), but that the singularities are reduced at most to a pole at s = 1 is not yet proved.

Definition 3.2.1. We say that f ∈ Cd has the ∗-property when f ⊗ χ is an entire function for all primitive χ (hence f is entire as well, since f = f ⊗ χ0 with q = 1).

The previous remarks show that there are elements with the ∗-property in Cd for d = 2, 3 (see Remark 3.2.2) and conjecturally for every d ≥ 2, but not every element of Cd satisfies the ∗-property, as the function ζ2(s) shows. However, there is a strong evidence, but no proof, that the elements of Cd with d ≥ 2 satisfy the ∗-property if they are not a product or Rankin-Selberg convolution of functions in some Cd0 (see Remark 3.2.4). The main result of this chapter is proving that the restriction d ≥ 2 is in fact a necessary condition for the ∗-property.

Theorem 3.2.1. Let f ∈ C1 satisfy the ∗-property. Then f is the constant function f(s) = 1.

∗ The class Cd appears to be related to the L -classes of Chapter 2 and to the Selberg class Sd (see [56] and Kaczorowski and Perelli [29]), but there are some important differ- f ences. Firstly, in Cd the kernel Φχ of the functional equation is not necessarily a product of Γ-factors; secondly, in Cd we assume a ”well-behaviour” of f ⊗ χ that probably is true for Sd as well, but f ⊗ χ does not necessarily belong to Sd. And, at last, in our arith- metical definition d is always an integer, while in the Selberg setting every positive real value is in principle possible for d, as a consequence of a different (analytic) definition. In all known cases the two definitions provide the same result: this circumstance revels that there are deep aspects of the theory that are not yet well understood. Kaczorowski and Perelli [29] proved that the Dirichlet L-functions L(s, κ) and their shifts are the only elements of S1, so it is natural to conjecture that these functions exhaust C1 as well. We are not able to prove this conjecture at present; however, our theorem agrees with this conjecture. The theorem is a consequence of Proposition 1.2.1 of Chapter 1 and of the following lemma. P −s P −s Lemma 3.2.1. Let f(s) = n ann ∈ C1 and g(s) = n bnn ∈ Cd for some d ≥ 2, and assume that f and g satisfy the ∗-property. Then

X 2n −A anbnη A x for every A > 0 x x 2

Proof of the theorem. From Lemma 3.2.1 we get −A (3.1) |anbn| < c(A)n for every A > 0. We write d Y −s −1 Y Y −s −1 f(s) = (1 − α(p)p ) , g(s) = (1 − βj(p)p ) . p p j=1

Given any prime p, we select a function g such that |βj(p)| = 1 for some j (this is always possible, for example in C2 with g normalized L-function associated with a holomorphic newform for SL2(Z)). Then the sequence bpk satisfies the hypothesis of Proposition 1.2.1, so there is a subsequence {bpkn } such that |bpkn | > c for some positive constant c and every n. The complete multiplicativity of an and (3.1) give

kn −knA |α(p)| c = |apkn |c ≤ |apkn bpkn | ≤ c(A)p ,

1 −A so |α(p)| ≤ (c(A)/c) kn p , and hence taking n → ∞, for any p and A we have |α(p)| ≤ −A p . Therefore α(p) = 0 for every p, and the result follows.  3.3. Proof of lemma 3.3.1. Preliminary identities. Let η be as in Lemma 3.2.1, X √ √ Z Y (x) := η(q/ x) ∼ x η(u)du q R and ∞ X n D(x) := a b η2 . n n x n=0 In order to study the asymptotic behaviour of D(x) and prove the lemma, we begin by performing the same transformations of Section 3 of [14], with some little changes to meet our situation. In particular, the decomposition of arm is now obvious by the complete multiplicativity. Moreover, the other arithmetical functions br(b), ct(c), dt(d), necessary for the decomposition of brn and to relax the constraints (m, t) = 1 and (n, t) = 1 respectively, are now defined by

X  (3.2.a) brn = br(b)bn0 , br(b)  r , bn0=n, b|rd−1 ( X bn if (n, t) = 1  (3.2.b) d (d)b 0 = , d (d)  t , t n 0 otherwise t dn0=n, d|td ( X am if (m, t) = 1  (3.2.c) c (c)a 0 = , c (c)  t . t m 0 otherwise t cm0=m, c|t

The existence of br(b) for d = 2 is proved in Duke and Iwaniec [13], and the general case is treated similarly; the existence of ct(c) and dt(d) is assured by the Euler product (in particular ct(c) = µ(c)ac, with µ the M¨obius function). 3.3. PROOF OF LEMMA 59

The result of these transformations is the following identity, which is analogous to (9) of [14]

X −1 X X (3.3) Y D(x) = φ(qt) arbr(b) ct(c)dt(d)· q,r,t (b,qt)=1 (cd,q)=1 b|rd−1 c|t, d|td X∗ X crm bdrn qrt  χ(cm)¯χ(bdn)a b h , , √ , m n x x x χ mod q m,n where  |x − y| h(x, y, z) := η(x)η(y) η(z) − η z has support in [1/2, 1] × [1/2, 1] × (0, 1] and P∗ is a sum on the primitive characters only.

Now we adapt to our case the argument in Section 4√ of [14], but we avoid using Kloosterman sums. Let ρ1 := cr/x, ρ2 := bdr/x, z := qrt/ x, h(u, v) := h(ρ1u, ρ2v, z) and X ∆(χ) := χ(m)¯χ(n)ambnh(m, n). m,n h(u, v) is a smooth function with compact support, vanishing in {|u| < 1/2ρ1} × {|v| < 1/2ρ2}, hence ZZ ∞ ˇ −s1 −s2 h(s1, s2) := h(u, v)u v dudv 0 ˇ s1−1 s2−1ˇ is entire in C × C. Moreover, the equality h(s1, s2) = ρ1 ρ2 h(s1, s2, z) holds with ZZ ∞ ˇ −s1 −s2 (3.4) h(s1, s2, z) := h(u, v, z)u v dudv , 0 therefore ZZ ∞ −s1 −s2 ˇ s1−1 s2−1 ρ1 ρ2 h(1 − s1, 1 − s2, z) = h(u, v)u v dudv . 0 Inversion of this Mellin integral gives −1 ZZ h ˇ −s1 −s2 (u, v) = 2 h(1 − s1, 1 − s2, z)(ρ1u) (ρ2v) ds1ds2 , 4π σ1,σ2>1 therefore −1ZZ ˇ −s1 −s2 ∆(χ) = 2 h(1 − s1, 1 − s2, z)(f ⊗ χ)(s1)(g ⊗ χ¯)(s2)ρ1 ρ2 ds1ds2 4π σ1,σ2>1 P −s P −s by the uniform convergence of n ann and n bnn in σ > 1 + . The functions f ⊗ χ and g ⊗ χ¯ are entire by the ∗-property and have a polynomial behaviour on vertical strips by the hypothesis on the functional equations. In the next section we prove that hˇ tends to zero on vertical lines faster than any power, so the 60 3. EXISTENCE OF A SINGULARITY FOR CERTAIN FUNCTIONS OF DEGREE 1

changes s1 → 1 − s1, s2 → 1 − s2 and the subsequent applications of Fubini and Cauchy theorems give −1 ZZ ˇ s1−1 s2−1 ∆(χ) = 2 h(s1, s2, z)(f ⊗ χ)(1 − s1)(g ⊗ χ¯)(1 − s2)ρ1 ρ2 ds1ds2 . 4π σ1,σ2>1 Now we use the functional equations and again introduce the Dirichlet series, thus getting q−(1+d)/2 X  m n qrt  ∆(χ) = χ¯(m)χ(n)¯a ¯b H , , √ , ρ ρ m n χ qρ qdρ x 1 2 m,n 1 2 where −1 ZZ ˇ f g −s1 −s2 (3.5) Hχ(u, v, z) := 2 h(s1, s2, z)Φχ(s1)Φχ¯(s2)u v ds1ds2 . 4π σ1,σ2>0

In the definition of Hχ we can allow every positive value for σ1 and σ2, by the hypothesis f g ˇ about Φχ and Φχ and the behaviour of h on vertical lines. Substituting this expression in (3.3) we obtain the final equality

2 X X X E (3.6) Y D(x) = x ar br(b)ct(c)dt(d) 2 , √ bcdr rt< x b|rd−1 c|t (b,t)=1 d|td where − 1+d X q 2 X∗ mx nx qrt  (3.7) E := a¯ ¯b χ(cnbdm)H , , √ , ϕ(qt) m n χ crq bdrqd x m,n,q χ mod q (bcdmn,q)=1 which is the analogue of (10) of [14].

|u−v|

3.3.2. Estimate of Hχ. We recall that h(u, v, z) = η(u)η(v) η(z) − η z has ˇ support in [1/2, 1] × [1/2, 1] × (0, 1], and the definitions of h(s1, s2, z) and Hχ(u, v, z) in (3.4) and (3.5). By partial integration we have, for every A, B ≥ 0

ZZ ∞ A−s1 B−s2 ˇ ∂h(u, v, z) u v h(s1, s2, z) = A B dudv . 0 ∂ u∂ v (s1 − A) ... (s1 − 1) (s2 − B) ... (s2 − 1)

A+B ∂h(u,v,z) Moreover, z ∂Au∂B v is uniformly bounded on its support, since it is a polynomial in z, η(i)(u), η(j)(v), η(k)(|u − v|/z). Hence the former relation gives the estimate ˇ −A−B −A −B (3.8) h(s1, s2, z)  z (1 + |s1|) (1 + |s2|) for every A, B ≥ 0

where the implied constant depend only on A, B, σ1, σ2, thus (3.8) is uniform on vertical lines. Therefore f g ZZ |Φ (s1)| |Φ (s )| −σ1 −σ2 −A−B χ χ¯ 2 Hχ(u, v, z)  u v z A B dt1dt2 , σ1,σ2>0 (1 + |s1|) (1 + |s2|) 3.3. PROOF OF LEMMA 61

the estimate being independent of the character χ if σ1 and σ2 are sufficiently large. f σ1 f B(σ2) Moreover, we have assumed that Φχ(s1)  |t| and Φχ(s2)  |t| for |t| > 1 and σi large, so ZZ +∞ −σ1 −σ2 −A−B σ1−A B(σ2)−B Hχ(u, v, z)  u v z (1 + |t1|) (1 + |t2|) dt1dt2 , −∞ where, by (3.8), we can suppose A and B sufficiently large to assure the convergence of the integral. Choosing A = σ1 + 1 +  and B = B(σ2) + 1 + , we have

σ1−B(σ2)−2− σ1+B(σ2)+2+ −σ1 −σ2 −σ1−B(σ2)−2− − 2 −σ2 2 − 2 Hχ(u, v, z) σ1,σ2 u v z = u v (uz )

for every large σ1, σ2, i.e., −A −D 2 −B˜ Hχ(u, v, z) A,D u v (uz ) for every large A, D > 0, for some B˜ = B˜(A, D) > 0 . Hence mx nx qrt  crq Abdrqd Dmx q2r2t2 −B˜ (3.9) H , , √  . χ crq bdrqd x A,D mx nx crq x In view of the support of h, H (u, v, z) is zero for z > 1, so we can greatly simplify χ √ x estimate (3.9) by assuming 0 < z ≤ 1, i.e., q ≤ rt . In fact crq cr x1/2 x−1/2 ≤ ≤ mx mx rt m by (3.2.c), bdrqd b d x(d−2)/2 x(d−2)/2 ≤ ≤ nx rd−1 td n n by (3.2.a) and (3.2.b), and mx q2r2t2  ≥ 1 crq x by (3.2.c). Thus (3.9) becomes

− A + d−2 D mx nx qrt  x 2 2 H , , √  for every A, D > 0. χ crq bdrqd x A,D mAnD Finally, with a suitable choice of D = D(A), we have mx nx qrt  x−A (3.10) H , , √  for every A > 0. χ crq bdrqd x A mAnA uniformly on χ. 3.3.3. Estimate of E. Estimate (3.10) is strong enough that we can bound E triv- ially, using the uniformity on χ and taking the absolute values in (3.7). Hence

1−d −A X q 2 X |am| X |bn| x (3.11) E  x−A  for every A > 1, A ϕ(qt) mA nA A t1− q m n where the q-series is convergent since we have assumed d ≥ 2, and the same holds for the m and n-series when A > 1. 62 3. EXISTENCE OF A SINGULARITY FOR CERTAIN FUNCTIONS OF DEGREE 1

 3.3.4. Proof of lemma. The bound (3.11), the trivial estimates ar, br(b)  r , ct(c),  dt(d)  t and b, c, d ≥ 1 give, when introduced in (3.6), that     2−AX r t X 2−AX r t 2+−A Y D(x) A x 2 1 A x 2 A x for every A > 1. √ r t √ r t rt≤ x b|rd−1 rt≤ x c|t d|td √ This completes the proof of Lemma 3.2.1, since Y  x.

3.4. Appendix

Writing f(s) = L(s, κ) with κ primitive character modulo q0, we want prove that f ∈ C1, so we have to study the functional equation of f ⊗ χ where χ is a primitive character modulo q. Let υ be the character modulo q1 (q1|q0q) that induces κχ, so that the identity f ⊗ χ = L(s, υ) Q (1 − υ(p)p−s) holds. It follows that f ⊗ χ satisfies the p|q0q functional equation

s+νυ s−1 2s−1 2s−1 Γ( ) 1 − υ(p)p −νυ 2 − 2 Y ¯ f ⊗ χ(1 − s) = i υq π 2 f ⊗ χ¯(s), 1 Γ( 1−s+νυ ) 1 − υ¯(p)p−s 2 p|q0q √ where νυ is the parity of υ and υ = τ(υ)/ q1 (phase of the Gauss sum). We write the functional equation selecting the following factors

2s−1 2 ˜ ¯ f ⊗ χ(1 − s) = q αυΨνυ (s)Ψ(κ, χ, s)f ⊗ χ¯(s), where

−νυ αυ : = i υ

2s−1 s+νυ q  2 Γ( ) Ψ (s) : = 0 2 νυ 1−s+νυ π Γ( 2 ) 2s−1 s−1 ˜  q1  2 Y 1 − υ(p)p Ψ(κ, χ, s) : = −s . q0q 1 − υ¯(p)p p|q0q

Here |αυ| = 1, Ψνυ (s) is an holomorphic function in σ > 0 that depends only of the parity of υ, with a |t|σ-behaviour on vertical lines by the Stirling formula, and Ψ(˜ κ, χ, s) is an holomorphic function in σ > 0, bounded on vertical strips but depending of the character χ. ˜ Verifying that f ∈ C1 means then proving that Ψ(κ, χ, s) is uniformly bounded on χ and t for large and fixed σ: we prove this for σ > 0. In fact

2σ−1 σ−1 2σ−1 σ−1 ˜  q1  2 Y 1 + p  1  2 Y 1 + p (3.12) |Ψ(κ, χ, s)| ≤ −σ ≤ −σ q0q 1 − p M 1 − p p|q0q p|M p-q1 3.4. APPENDIX 63

σ−1 1+p q0q since −σ > 1 and M := is an integer. If we assume σ ≥ 1, (3.12) implies that 1−p q1 2σ−1 1−σ  1  2 Y Y 1 + p c() (3.13) |Ψ(˜ κ, χ, s)| ≤ pσ−1 ≤ , M 1 − p−σ M 1/2− p|M p|M where we have used (1 + p1−σ)/(1 − p−σ) ≤ 4 for all p. Estimate (3.13) is particularly interesting since it is uniform on the character κ as well. The bound (3.13) holds in σ > 1 and it is sufficient to prove that L(s, κ) ∈ C1, but we further observe that an estimate, uniform in χ but not in κ, is still possible for 0 < σ; in 2 2 fact we will prove that M| MCD(q0, q ), thus from (3.12) we have Y 1 + pσ−1 |Ψ(˜ κ, χ, s)| ≤ max(1, q1−2σ) 0 1 − p−σ p|q0 that is independent of χ. 2 2 Q ap Q bp Q cp For a proof of M| MCD(q0, q ), let q0 = p p , q = p p , q1 = p p be the p-parts Q a Q Q a of the moduli and κ = p κp p , χ = p χpbp and υ = p υp p be the p-parts of the characters. Then κpap , χpbp and υpcp are primitive and υpcp induces κpap χpbp . We prove that if ap 6= bp, then cp = max(ap, bp). In fact let ap < bp and, by contradiction, cp < bp. ap ap cp bp κ¯pap is a character modulo p soκ ¯pap υpcp is a character modulo max(p , p ) < p , hence bp it induces a character mod p that cannot be primitive. This is a contradiction since χpbp is the induced character. It follows that Y pap pbp Y pap pbp Y pbp Y M = = = pap+bp−cp , pcp pcp pcp p p|q0 p-q0 p|q0 but ap 6= bp implies ap + bp − cp = min(ap, bp) and ap = bp implies ap + bp − cp ≤ 2ap, hence 2 2 M|q0. In a similar way we prove that M|q too.

CHAPTER 4

About the Selberg class Sd, 0 ≤ d ≤ 1

4.1. Introduction and results In the analysis of Dirichlet series with arithmetic significance, Selberg [56] introduced an axiomatically defined class S of functions and formulated some conjectures about it that have many astonishing consequences (see Conrey-Ghosh [9] and Murty [54]). We refer to Kaczorowski-Perelli [31] for a survey of the properties of the class S. From the study of the structure of S for small value of the degree (see the following definition) that has been performed by Kaczorowski and Perelli in [29], it is clear that also the more general class S] ⊃ S is a very interesting object. We recall its definition: ] P∞ −s (i) (Dirichlet series) Every f(s) ∈ S is a Dirichlet series f(s) = n=1 a(n)n , absolutely convergent for σ > 1. (ii) (Analytic continuation) There exists m ∈ N such that (s − 1)mf(s) is an entire function of finite order. (iii) (Functional equation) Every f(s) ∈ S] satisfies a functional equation of type Ψ(s) = ωΨ(1¯ − s), where r s Y Ψ(s) := Q Γ(λjs + µj)f(s), j=1 ¯ with Q > 0, λj > 0, <µj ≥ 0 and |ω| = 1. Here f(s) := f(¯s). ] P ] ] For f(s) ∈ S , the degree is defined as df := 2 j λj and Sd := {f(s) ∈ S , df = d}. The arguments in Richert [55], Bochner [4] and Conrey-Ghosh [9] prove the following

] Theorem. (Richert, Bochner, Conrey-Ghosh) Sd = ∅ when 0 < d < 1. Moreover, if ] 2 P −s f(s) ∈ S0, then Q ∈ N, f(s) is a Dirichlet polynomial f(s) = n|Q2 a(n)n , and the n Q2 functional equation is equivalent to the relation a(n) = ω Q a¯( n ). Actually, the first part of this theorem has been conjectured by Selberg [56], and first proved by Richert and Bochner, independently. In recent years this result has been redis- covered by Conrey-Ghosh which have also proved that S0 (see e.g. [31] for its definition) is reduced to the constant function f(s) = 1. The second part of the theorem, i.e., the ] finer properties of S0 is due to Kaczorowski and Perelli [29]. We present two conceptually simpler proofs of the above theorem, based on a standard approach. The idea for the first proof is to employ a well known integral formula for the m-th coefficient a(m) of f(s), so that a contour integral estimate allows us to prove the P result. The second proof uses in a new way the classical Perron formula for n≤x a(n). 65 66 4. ABOUT THE SELBERG CLASS Sd, 0 ≤ d ≤ 1

It has been pointed to our attention that our first proof was already been used by Richert ([55], Hilfssatz 11): a paper we did not know when [42] was submitted for publica- ] tion. Nevertheless, we employ this technique to deduce also the claim about S0 contained into the theorem. The second approach has two interesting corollaries: Corollary 4.1.1. (Kaczorowski-Perelli) The Ramanujan conjecture a(n)  n is ] true for every f(s) ∈ Sd, 0 ≤ d ≤ 1. ] (d−1)/2+ Corollary 4.1.2. For every f(s) ∈ Sd with d ≥ 0, we have a(n)  n . Corollary 4.1.1 has already been proved in [29] as a consequence of the complete ] determination of S1 given in that paper. However, our proof of this corollary is direct and very simple. ] Corollary 4.1.2 is non-trivial only for d<3, and might be useful in the study of Sd in this range. It follows also from theorem 3.1 in Chandrasekharan and Narasimhan [8], where an P ρ evaluation of n≤x a(n)(x − n) is provided, but our iterative approach (see the following definition of the sequence {cm}) makes its proof easier. At last, Corollary 4.1.2 should be compared with a result by Duke-Iwaniec [12]. In fact, as a consequence of theorem 1 χ P∞ −s ] in [12] we have that if f (s) := n=1 a(n)χ(n)n belongs to S for a sufficiently large number of primitive characters χ and, moreover, its functional equation satisfies some d−1 + compatibility conditions, then the stronger bound a(n)  n d+1 holds. 4.2. First proof We write the functional equation in the form f(1 − s) = Φ(s)f¯(s), where

Y Γ(λjs +µ ¯j) Φ(s) := ωQ2s−1 . Γ(λ (1 − s) + µ ) j j j For γ > 0 and any fixed integer m ≥ 1 we have the well known formula 1 Z 1+γ+iT (4.1) a(m) = lim f(s)msds . T →+∞ 2iT 1+γ−iT By the residue theorem and the functional equation, the integral in (4.1) is equal to Z −γ−iT Z −γ+iT Z 1+γ+iT (4.2) mP (ln m) + f(s)msds − f(s)msds + Φ(s)f¯(s)m1−sds , 1+γ−iT 1+γ+iT 1+γ−iT where P (x) is a certain polynomial. d (1+2γ) The functional equation gives the estimate f(−γ + it)  (|t| + 1) 2 , and hence d (1+2γ) f(s)  T 2 uniformly in the strip −γ < σ < 1+γ by the Phragm´en-Lindel¨oftheorem. d (1+2γ) Therefore, the horizontal integrals in (4.2) are Om(T 2 ). The third integral in (4.2) is ∞ Z 1+γ+iT i X a¯(n) Z T (4.3) Φ(s)f¯(s)m1−sds = Φ(1 + γ + it)(nm)−itdt mγ n1+γ 1+γ−iT n=1 −T and we treat this integral in different ways for d > 0 and d = 0. 4.3. SECOND PROOF 67

4.2.1. Case d > 0. We use Stirling’s formula for the Γ function to select the main contribution when T 1. The van der Corput method (stationary phase, see Lemma 4.5 d (1+2γ)+ 1 of Titchmarsh [63]) gives the estimate O(T 2 2 ), uniformly in nm, for the integral in the right side of (4.3). From (4.1), (4.2) and the latter estimate we get

d (1+2γ)− 1 a(m) m lim T 2 2 T →∞ and the first part of the theorem follows since the exponent is negative for d < 1 and γ sufficiently small. 4.2.2. Case d = 0. In this case Φ(s) = ωQ2s−1 and hence (4.3) becomes ∞ Q1+2γ X a¯(n) Z T  Q2 it m Q2  (4.4) iω dt = 2iT ω a¯ + O (1) mγ n1+γ nm Q m m n=1 −T Q2 2 2 where obviously the terma ¯( m ) is present only if Q ∈ N and m|Q . Hence the second part of the theorem follow from (4.1), (4.2) and (4.4).

4.3. Second proof We assume d > 0 and use the Perron integral formula X 1 Z 1+γ+iT xs x1+γ+A  a(n) = f(s) ds + O 2πi s T n≤x 1+γ−iT where γ > 0 and the positive constant A is such that a(n)  nA. Then we use the residue theorem on the path 1 + γ − iT → −γ − iT → −γ + iT → 1 + γ + iT as in the previous section; a term of type xP (ln x) appears from the possible pole at s = 1 of f(s), and the constant f(0) as residue at s = 0. We estimate the horizontal integrals by the functional equation and the Phragm´en- Lindel¨oftheorem, and the vertical one by the van der Corput method as in the previous section, obtaining 1+γ+A d (1+2γ)− 1 X x  T 2 2  a(n) = xP (ln x) + f(0) + O + O . T xγ n≤x For T (d+1)/2  x1+A and γ =  arbitrarily small we get

X  (1+A) d−1 + a(n) = xP (ln x) + O x d+1 , n≤x and hence

(1+A) d−1 + (4.5) a(n) A n d+1 by subtraction. For d < 1 the exponent in (4.5) is negative, and we may choose A arbitrarily large, −k so that a(n) k n for any k > 0. Then f(s) is uniformly bounded in every right half-plane, a contradiction when d > 0 since the functional equation implies in this case that the Lindel¨offunction µf (σ) is positive for σ < 0. This proves the first part of the 68 4. ABOUT THE SELBERG CLASS Sd, 0 ≤ d ≤ 1

theorem, and the second one follows easily from the functional equation in the case d = 0.

For d ≤ 1, (4.5) gives a(n)  n, i.e., Corollary 4.1.1. d−1 We define the sequence {cm} by c1 = A and cm+1 = (1 + cm) d+1 + . Hence cm → (d − 1)/2 + (d + 1)/2 as m → ∞. We have already proved that a(n)  nc1 implies c2 cm a(n)  n , so a recursion gives a(n) m n for every m. Since there exists M such d−1 + d+2 that cM < (d − 1)/2 + (d + 2)/2, we get a(n)  n 2 2 , i.e., Corollary 4.1.2. Bibliography

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