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provided by Elsevier - Publisher Connector JOURNAL OF NUMBER THEORY 15, 14-24 (1982)

Elementary Proof of the Transformation Formula for Lambert Involving Generalized Dedekind Sums

TOM M. APOSTOL

Department of Mathematics, California Institute qf Technologv. Pasadena, California 91 I25

Communicated bv 0. Taussky Todd

Received September 24. 1980

An elementary proof is given of the author’s transformation formula for the Lambert series G,(x) = Cz-, n I’.Y”/( 1 -x”) relating G,(e’“‘) to G,(e’““), where p > I is an odd integer and Ar = (ar + b)/(cr + d) is a general modular substitution. The method extends Sczech’s argument for treating Dedekind’s fun&ion log q(r) = nir/l2 - G,(e”“). and uses Carlitz’s formula expressing generalized Dedekind sums in terms of Eulerian functions.

1. INTRODUCTION

Let As = (ar + b)/(cs $ d) be a unirnodular substitution, where a, 6, c, d I are integers with ad - bc = 1. A transformation formula for the Larnbert series

G,(x) = 2 n -px”/( I -x”) (1x1 < 1) n = I

relating Gp(eZni’) to Gp(eZniil’) was derived for odd integer p > 1 by the author [ 11, using a Mellin-transform technique developed by Rademacher [ 19 1 in treating Dedekind’s function log ~(5) = nir/l2 - G,(e’““). The formula, which is stated below in Section 3. contains generalized Dedekind sums. The result for the special substitution AT = -l/r. in which the generalized Dedekind sums do not appear, had been found earlier by Guinand 115, Theorem 9(iv)]. Subsequent proofs of the general formula have been given by Mikolis [ 18 1, Iseki I17 1, Apostol (31, Glaeske ( 131, Bodendiek (7 1. Bodendiek and Halbritter 18 1, and Berndt 14.5 1. None of these proofs can be considered elementary since they employ techniques such as residue calculus.

00?2~314x:82:040014~l1%02.00;0 Copyright Cl 1982 by Academic Press, Inc. All rights of reproduction in any form reserved. TRANSFORMATION FORMULA FORLAMBERT SERIES 15

Mellin transforms, the Poisson summation formula, and functional equations of various zeta functions. Recently Sczech [21] has given an elementary proof of Dedekind’s transformation formula for log q(t) based on an idea of Hurwitz [ 161 which relates the two values obtained when the order of summation is reversed in a conditionally convergent double . This paper shows that Sczech’s argument can be extended to treat the more general case of G,(x) in an elementary way. An interesting by-product of the proof is the role played by the Eulerian functions H,(x), discussed below in Section 6.

2. NOTATION

We write At = (ar + b)/(cs + d), w h ere ad - bc = 1, c > 0, and Im(r) > 0. As usual, i(p) = C,“, n -p denotes the . The generalized Dedekind sums S,(a, c) are defined for 0 < r < p + 1 by the equation

S,(a, c) = \‘ PAP/C) pp + I ,@P/C). (1) umodc where (a, c) = 1, the summation is over a complete residue system mod c, and P,(x) = B,(x - [xl) is the periodic extension of the Bernoulli polynomial given by the Fourier series

n! 2lrik.x P,(,~) = - (2ni)” y” e k” for all real x if n > 1, and for x # integer if n = 1. Here we have adopted the Eisenstein notation 123, P. 61

\‘ and L“ = \_I, .

For n > 1 the numbers P,,(O)are the Bernoulli numbers, denoted by B,. and (2) implies that B, = 0 for odd n > 1. and B, = -2n!(2~i)~” i(n) for even II > 2. We also have B, = -1. Carlitz [9] has shown that the sums S,(a, c) can be expressed in terms of the rational functions H,(x) introduced by Euler [ 11 ] through the

1 -x -= q- H,(x)5 et-x n- = n H! 16 TOM M. APOSTOL

These functions are described more fully in Section 6. The version of Carlitz’s formula that will be needed in our elementary proof states that

B Sr(-a, c) = pyBr + r(p+ cp1 -Y) 7 Hp-,Kvf-di”? , (4) &;,, (r” - l)(C”” - 1) where c= ezxi”. (See 19, p. 521 J, where Carlitz uses a slightly different notation. Our sums S,.-u, c) are the same as Carlitz’s sums c,(-d, c) where ad= 1 (mod c).) We also note that

3. THE TRANSFORMATIONFORMULA

THEOREM. If p > 1 is an odd integer we have

Gp(eZni’) = (CT+ d)pm’ G,(e 2rri.4r)- i<(p){ 1 - (CT+ d)P- ’ } + A,(t), (5) where

A,(r) = (CT+ d)P--r S,(-a. c). (6)

Note. Because of an error in computing residues, the term on the right involving c(p) is missing in the original derivation of (5) in [ 11. ProoJ Since

we have

Gp(eZni’) - (cr + d)Pm ’ G,(e 2niAT)= -fc(p)( 1 - (cr + d)P-m’ } + A,(r), where

i T A,(*) = 5 -I n PP(cot mu - (CT+ d)P-’ cot xnAs}.

Thus, the proof amounts to summing this series of cotangents in the closed form given by (6). (Note. For the special transformation At = -l/r this series of cotangents was also summed by Berndt [6] using residue calculus.) TRANSFORMATIONFORMULAFORLAMBERT SERIES 17

Since cot is an odd function and p is odd, we can write

A,(r) = + x: KP(-cot 7cnr + (CT + Q-’ cot 7x47}. n

We rearrange this absolutely convergent series, summing first over all n belonging to a fixed residue class mod c, say n = ,D (mod c), and then over the c different residue classes ,Umod c. This gives

Ap(5)=y;= -\’ ,e\” =R,+R, (7) n u mod c n=p(mod c) where

Here notation such as n -p(c) means that the sum is extended over all integers n Z,B (mod c). We treat the sums R, and R separately.

4. EVALUATION OF THE SUM R,

We replace the cotangent terms by the series

ncot7rW=\‘,(m+w)-1. (8) m

Since cot xw has period 1 and

1 A,,: - (9) C c(cr + d) we have, for n = 0 (mod c>, cot rcnAs = cot i and hence n -I iTCOtmL4S=~,m- \’ c(cz + d) m 1 c(cr + d) 1 = ;;;’ mc(cs + d) - n and 1 -7tC0t71nr=~e-----= l 1’ m m-m fern-n+k I8 TOM M.APOSTOL where k is any integer independent of m. Thus,

c(cr + d)” t mc(cr + d) - n i ’ Now we replace n by nc and then choose k = -nd to obtain

1 1 + (cr + d)P R,=p \“ \‘ n-P 47ticp 7’;;;” 1 m - n(cr + d) m(cr + d) - n ’

The terms with m = 0 contribute

1 \“ 1 + (cr + d)P npp 4nicP 7;” i -n(cr + d) -n I

=-r(p+l) ’ +(cT+d)P (10) 2niP i cr + d 1. The remaining contribution to R, is

1 1 \“ \“ npP (cr + d)’ 1 4nicP 7;’ be i m - n(cr + d) ’ m(cr + d) - n i ’ Since now m and n range over the same set of values we can interchange m and n in the first double sum and this contribution becomes

1 ~ 1 \” \“ (cr + d)P 4nicP ye;;;” 1m”(n - m(cr + d)) ’ nP(m(cr + d) - n) t ’ Using the identity xp - JJ~= (X - J,) ce , .I?‘~~ ’ we see that the expression in brackets is equal to

np - m”(cr + d)P \“’ (cr f d)P-’ n’m’mPm r, npmp(n - m(cr + d)) = r~, and when this is summed over m and n we get

since \ .I -6’ n ’ = 2c(k) II TRANSFORMATION FORMULA FOR LAMBERT SERIES 19 if k is even, and =0 if k is odd. Combining this with (10) we obtain

1 R,=--- (CT+ djP-’ i(p + 1 - r) i(r) 7Ccp r: I IreVeIl) _ mJ + 1) & t (cr t qp . (11) 27ticp i i Expressing the zeta functions in terms of Bernoulli numbers we find

l (y $ (“T 1) (CS+d)p-rBp+,-rBr, (12) R”= 2(p + l)! where now the restriction r even is not needed since B, = 0 for odd r > 1.

5. EVALUATION OF THE SUM R

Next we treat the sum

Rx; x \‘;n-” {-cot 7Tnr+ (CT+ d)P-’ cot nt?As}. LlfO(C)n=p(c) First we modify the summand involving cot rrnt. Since ad - bc = 1 the numbers a~ and ,L run through the same set of residue classes mod (7, so

where k = a~ - ,u. Using this with f(n) = n Pp cot 71117 in R we obtain

(CT+ d)“- ’ cot mAr cot rr(n + k)r 1 Rx; \‘ \‘: U&O(C)n=u(c) 1 np (n+k)P \’ (13) Once again we use the cotangent series (8) together with (9) and take into account the congruence n -p (mod c) to obtain

n7t n I rccot mAs= rrcot ?% c C(CT+ d) C(CTt d‘;

= c \‘e (mc +pua - n/(cr + d)) ’ m =c(ct+d) \‘e (m(cz+d)-rzm’. t?l-OU(C) 20 TOM M. APOSTOL

For the other cotangent term we write

-7c cot 77(n + k)t = \‘e (m - r(n + k)). ’ m

=\‘,(m-r(n+k)+r)y’, (14) m where r is any integer independent of m. We choose r = (qu -k - d(n + k))/c. This is an integer because k = a,~ -,u and

up-k-d(n+k)=p-d(n+k)=p(l-ad)+d(p-n) = -,ubc + d(p - n) E 0 (mod c), since n = ,U (mod c) and ad - bc = 1. Therefore (14) becomes

-n cot rc(n + k)r = c \‘<> (mc - ct(n + k) t rc) ’ m =cx:,(mc+ap-k-(crtd)(ntk))~’ m c \‘ (m-k-(u+d)(n+k)) ‘. I?l=OM(C) Thus (13) becomes

1 (CT+ dY ’ (n t k)P(m - k - (cr t d)(n + k)) t nP(m(cr t d) - n) ’ Since n E,L (mod c) if and only if n $ k -,B t k s up (mod c) we can interchange m and (n + k) in the summand without altering the sum. We do this in the first term to get

mp(c5 + d)P - np m(ct + d) - n

where (15) TRANSFORMATION FORMULA FOR LAMBERT SERIES 21

In the next section we evaluate the sums F,(s) in terms of the Euler functions and then show that

c 1 F,(P+ 1 -r>For(r) W&O(C)

so that

R = (2ni)P + p + l (c7 + d)PF’S r(-a 3c) 2(p + l)! ,-I i r 1 _ (274” <- P + 1 (CT+ d)p-lBP+l--‘Br 2(p+ I)! ,Tj i r 1 cp .

From (12) we see that the last sum, when extended over the range r = 0 to r= p+ 1, is -R,, so

R$R,=

B ,B + (CT + gp __ ;‘+(cT+d)- $p i (27q “\J p + 1 (CT + d)P-’ Sr(-u, c) = 2(p + l)! r:O t r ) since cmPBp+ , = S&a. c) = S,, ,(-a, c). Using this in (7) we obtain (6).

6. EVALUATION OF F,(~)IN TERMS OF EULER FUNCTIONS

It remains to prove (16). We do this with the help of the Euler functions H,(x) generated by (3), where x# 1 but is otherwise arbitrary. The H,(x) can also be defined by the recursion relation

<- n ffA4 = Xff”(X) for ~21, H,(x) = 1. r=o4 r ) For n > 1 they are rational functions of x given by

A,(x) ffn(x) = (x _ 1)” 3 22 TOM M. APOSTOL where A,(x) is a polynomial in x of degree n - 1,

A,(x) = -;’ A,, .$ ‘. r-l The coefficients A,,,. called Eulerian numbers, are integers satisfying the recursion relation

A n.k =kAn-,.k +(n-k+ l)A,-,,k-,.

The first few polynomials are

A,(x) = 1, A,(x) = 1 fx, A,(x) = 1 + 4x + x’, A‘,(x) = 1 + 1 lx + 1 lx2 +x3, A,(x) = 1 + 26x + 66x’ + 26x3 +x4, and they illustrate the symmetry property A,,, = A,.,, , or. These polynomials have a long and interesting history. For example, if 1x1 < 1 we have

They also occur in the evaluation of the Lerch zeta function at the nonpositive integers (see 121) and in combinatorial problems. Properties of the polynomials A,(x) and the associated rational functions H,(x) have been studied by many authors including Frobenius [ 121, Worpitzky (241, Carlitz [lo], Vandiver [22], and Riordan [20]. Professor Berndt informs me that Ramanujan also studied these polynomials in Chapter 5 of his second notebook. Frobenius has related the functions H,(x) with the Bernoulli functions. If p is a kth root of unity, p # 1, it is easily verified that

For p = 1 we have the corresponding relation

=B,

Hence if [ = e’“‘/’ and s is an integer, s > 1, we have

if af 0 (mode) rmodc (17) = c’ +Br if u-0 (mode). TRANSFORMATION FORMULA FORLAMBERT SERIES 23

We now use (17) to express the sum F,(s) in (15) in terms of the H, ~, for any integer s > 1. This result is of independent interest so we state it as a lemma.

LEMMA. Let [ = e2niJc. Then for arbitrary integers a and s, s > 1, we have

if a f 0 (mod c). (18) 2ni ‘B, =- zy a = 0 (mod c). C-1C sl Proof: To incorporate the congruence condition m = a (mod c) we introduce the geometric sum

Znirtm-o)lc = 0 xe if m&a (mode) rmodc

=C if m = a (mod c). Then

rmodc

The last sum can be evaluated by (17). giving us (18). Now we use the lemma to obtain

C \‘ F,(P+ 1 -r)F,,(r) UfOlC) = (27d)p+’ P+ 1 (p+ l-r)- ,, q.-,(C”) ff- ,(i”“) (p+ l)! c r 1 CP u *o(c)- (C-W- l)(<-““- 1). Because of (4) this gives (16) and completes the proof of (5).

REFERENCES

1. T. M. APOSTOL, Generalized Dedekind sums and transformation formulae of certain Lambert series, Duke Math. J. 17 (1950), 147-157. 2. T. M. APOSTOL, On the Lerch zeta function, Pacific /. M&h. 1 (195 1). 161-167. 3. T. M. APOSTOL. A short proof of ShB Iseki’s functional equation. Proc. Amer. Math. Sot. IS (1964). 618-622. 24 TOM M. APOSTOL

4. B. C. BERNDT. Generalized Dedekind eta-functions and generalized Dedekind sums. Trans. Amer. Math. Sot. 178 (1973). 495-508. 5. B. C. BERNDT, Generalized Eisenstein series and modified Dedekind sums, J. Reine Angewl. Math. 272 (1975), 182-193. 6. B. C. BERNDT. Dedekind sums and a paper of G. H. Hardy. J. London Math. Sot. 13(2) (1976). 129-137. 7. R. BODENDIEK. ober verschiedene Methoden zur Bestimmung der Transfor- mationsformeln der achten Wurzeln der Integralmoduln k’(r) und k”(r). ihrer Logarithmen sowie gewisser Lambertscher Reihen bei beliebigen Modulsubstitutionen. Dissertation, Kb;!n 1968. 8. R. BODENDIEK AND U. HALBRI~ER. uber die Transformationsforme! von log q(r) und gewisser Lambertscher Reihen. Abh. Math. Sem. Units. Hamburg 38 (1972). 147-167. 9. L. CARLITZ. Some theorems on generalized Dedekind sums, Pacific J. Math. 3 (1953). 513-522. IO. L. CARLITZ, Eulerian numbers and polynomials. Math. Mag. 32 (!958/3959). 247-260. 11. L. EULER, Institutiones calculi differentialis. (II), pp. 487-491, Petrograd. 1755. 12. G. FROBENIUS. uber die Bernoullischen Zahlen und die Eulerschen Polynome. Sitzungsber. Akad. Wissensch. Berlin 2 (1910). 809-847. 13. H.-J. GLAESKE. uber die Modultransformation einer Halbgitterfunktion. Arch. Math. (Basel) 17 (1966). 438-442. 14. A. P. GUINAND, On Poisson’s summation formula, Ann. of Math. 42 (1941). 59 I-603. 15. A. P. GUINAND, Functional equations and self-reciprocal functions connected with Lambert series. Quarf. J. Math. Oxford Ser. 15 (1944), I I-23. 16. A. HURWITZ. ober die Theorie der elliptischen Modulfunktionen. Math. Ann. 58 (1904). 343-360. 17. S. ISEKI, The transformation formula for the Dedekind modular function and related functional equations. Duke Math. 1. 24 (1957). 653-662. 18. J. MIKOLAS, uber gewisse Lambertsche Reihen. I. Verallgemeinerung der Modulfunktion q(r) und ihrer Dedekindschen Transformationsformel. Math. 2. 68 (1957), 100-I IO. 19. H. RADEMACHER, Zur Theorie der Modulfunktionen. J. Reine Anger. Math. 167 (1932). 312-336. 20. J. RIORDAN. Triangular permutation numbers, Proc. Amer. Math. Sot. 2 (195 1). 429-432. 2 1. R. SCZECH. Ein einfacher Beweis der Transformationsforme! fur log q(z), Math. Ann. 237 (1978). 161-166. 22. H. S. VANDIVER, An arithmetic theory of the Bernoulli numbers. Trans. Amer. Math. Sot. 51 (1942). 502-531. 23. A. WEIL. “Elliptic Functions According to Eisenstein and Kronecker.” Ergebnisse der Mathematik Band 88, Springer-Verlag. Berlin/Heidelberg/New York. 1976. 24. J. WORPITZKY. Studien fiber die Bernoullischen und Eulerschen Zahlen. 1. Roine Angew. Math. 94 (1883). 203-232.