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On the Values of a Class of Dirichlet Series at Rational Arguments

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On the Values of a Class of Dirichlet Series at Rational Arguments PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 138, Number 4, April 2010, Pages 1223–1230 S 0002-9939(09)10171-5 Article electronically published on December 4, 2009 ON THE VALUES OF A CLASS OF DIRICHLET SERIES AT RATIONAL ARGUMENTS K. CHAKRABORTY, S. KANEMITSU, AND H.-L. LI (Communicated by Ken Ono) Dedicated to Professor Eiichi Bannai on his sixtieth birthday, with great respect and friendship Abstract. In this paper we shall prove that the combination of the gen- eral distribution property and the functional equation for the Lipschitz-Lerch transcendent capture the whole spectrum of deeper results on the relations be- tween the values at rational arguments of functions of a class of zeta-functions. By Theorem 1 and its corollaries, we can cover all the previous results in a rather simple and lucid way. By considering the limiting cases, we can also deduce new striking identities for Milnor’s gamma functions, among which is the Gauss second formula for the digamma function. 1. Introduction and the finite value case Srivastava [S2] (cf. also Srivastava and Choi [SC], pp. 336–344) gives two simpler proofs of interesting results of Cvijovi´c and Klinowski [CK1] on the finite expression for the values of a Bernoulli polynomial at rational arguments in terms of the values of the Hurwitz zeta-function at integer arguments with rational values for the perturbation. The first proof uses the Fourier series for the periodic Bernoulli polynomial and the formula for the decomposition into residue classes. The second proof uses the functional equation for the partial zeta-function q p 2Γ (s) πs 2πap a (1.1) ζ 1 − s, = cos − ζ s, q (2πq)s 2 q q a=1 ([SC, (8), p. 90], [Ap]) to be restated as (1.11) below, where ∞ 1 (1.2) ζ(s, a)= ,σ>1, (n + a)s n=0 is the Hurwitz zeta-function. We introduce the Lipschitz-Lerch transcendent ∞ e2πinz (1.3) L(s, z, a)= , (n + a)s n=0 Received by the editors December 28, 2008, and, in revised form, August 18, 2009. 2010 Mathematics Subject Classification. Primary 11M35; Secondary 33B15. Key words and phrases. Lipschitz-Lerch transcendent, Hurwitz zeta function, polylogarithm function, Gauss’ second formula, Milnor’s gamma function. c 2009 American Mathematical Society Reverts to public domain 28 years from publication 1223 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1224 K. CHAKRABORTY, S. KANEMITSU, AND H.-L. LI absolutely convergent for σ>1ifz = x ∈ R or s ∈ C with Im z>0, which generalizes not only the Hurwitz zeta-function (1.4) ζ(s, a)=L(s, 1,a) but also the polylogarithm function (or the Lerch zeta-function) ∞ e2πinx (1.5) l (x)=L(s, x, 1) = ,σ>1. s ns n=1 The most general distribution property of the Hurwitz-Lerch transcendent (which follows from the principle of decomposition into residue classes used in Srivastava’s first method), q−1 a + j (1.6) L(z,s,a)=q−s L qz, s, e2πijz, q j=0 ([SC, (15), p. 339]) coupled with the functional equation for the Lipschitz-Lerch transcendent ([SC, (29), p. 125]), (1.7) Γ(s) πi( 1 s−2ax) −πi( 1 s−2a(1−x)) L(x, 1 − s, a)= e 2 L(−a, s, x)+e 2 L(a, s, 1 − x) , (2π)s captures the whole scene and gives an immediate proof of new limiting relations as well as all these types of linear combination expressions for a class of zeta-functions and as a special case for polynomials. Theorem 1. For 1 <q∈ Z with 0 <x<1 and Re a>0, one has the following identity: q−1 − −s a + j 2πijx Γ(1 s) (1.8) q L qx, s, e = L(x, s, a)= − q (2π)1 s j=0 πi( 1 (1−s)−2ax) −πi( 1 (1−s)−2a(1−x)) × e 2 L(−a, 1 − s, x)+e 2 L(a, 1 − s, 1 − x) except at the singularities of L, in which case, the identity is to mean the limit. Specializing the parameter a suitably, Theorem 1 reads: Corollary 1. For 1 <q∈ Z and q>p∈ N, we have the identity q −s 2πi a p a p q e q ζ s, = l q s q a=1 Γ(1− s) − πis p πis p (1.9) = i − e 2 ζ 1 − s, − e 2 ζ 1 − s, 1 − , (2π)1 s q q where, for s =0or s =1, the identity is to mean the limit as s → 0 or s → 1. We shall show that (1.9) and its counterpart for the polylogarithm function already entail not only all the previous results but also new non-trivial corollaries. As an example, we note that the trigonometric series (ν>1) ∞ ∞ cos(2k +1)πα sin(2k +1)πα (1.10) C (πα)= ,S(πα)= , ν (2k +1)ν ν (2k +1)ν k=0 k=0 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use DIRICHLET SERIES AT RATIONAL ARGUMENTS 1225 introduced by Srivastava and Choi ([SC, (1), p. 341]), may be thought of as the real and imaginary parts of the odd part of α l − 2−sl (α) . s 2 s Hence by the general distribution property (1.6) one has p p p p C π + iS π = l − 2−sl s q s q s 2q s q q − − −s 2j 1 2πi 2j 1 p =(2q) ζ s, e 2q . 2q j=1 This proves [SC, (8), (9), p. 342], which are equivalent to Cvijovi´c-Klinowski results [CK2] (cf. [S1]). Also note that in view of 2ν−1 − ν−1 (2π) ( 1) α −(2ν−1) C − (πα)= Cl − − 2 Cl − (α) 2ν 1 (2ν − 1)! 2ν 1 2 2ν 1 and 2ν − ν (2π) ( 1) α −2ν S (πα)= Cl − 2 Cl − (α) , 2ν (2ν)! 2ν 2 2ν 1 the statement by Srivastava-Choi about the slow convergence of the series is jus- tified. Here Clν (α) denotes the Clausen function (log-sine integrals; cf. [Le], [Y], [K]) defined as ∞ sin kα , if 2|n, Cl (α)= k=1 kν ν ∞ cos kα | k=1 kν , if 2 n. Corollary 2 ([Ap, Theorem 128, p. 261], [SC, (8), p. 90]). For all s =0 , 1, we have q p 2Γ (s) πs 2πap a (1.11) ζ 1 − s, = cos − ζ s, , q (2πq)s 2 q q a=1 and for n>1, 0 ≤ p ≤ q, q p 2n! 2πjp nπ j (1.12) B = − cos − ζ n, . n q (2πq)n q 2 q j=1 − πis Proof. To prove (1.11) we sum two equalities: one is e 2 times (1.9), and the πis other is e 2 times (1.9) with p replaced by q − p.Thenweget q −s − πs + 2πap i πs − 2πap i a q e( 2 q ) + e( 2 q ) ζ s, q a=1 − Γ(1 s) −πis πis p = i − e − e ζ 1 − s, (2π)1 s q License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1226 K. CHAKRABORTY, S. KANEMITSU, AND H.-L. LI or q πs 2πap a 2q−s cos − ζ s, 2 q q a=1 2Γ (1 − s)sinπs a 2π a = − ζ s, = − ζ s, , (2π)1 s q (2π)1 s Γ(s) q whence (1.11) follows upon using the functional equation for the gamma function. Now (1.12) follows from (1.11) in view of the formula (1.13) −nζ(1 − n, x)=Bn(x). Remark 1. (i) We note that the above deduction of (1.11) amounts to applying the Hurwitz formula (cf. (1.7)) Γ(s) − πis πis (1.14) ζ (1 − s, x)= e 2 l (x)+e 2 l (1 − x) . (2π)s s s We also note that the reciprocal relation for the gamma function is equivalent to the Hurwitz formula, while (1.13) is its consequence (cf. [Vista, Chapter 5]). (ii) As is noted in the first proof of [SC, Theorem 6.3, p. 336], it is rather simple to deduce the corresponding results on Euler polynomials from (1.12) by means of the relation 2 x +1 x (1.15) E (x)= B − B . n n +1 n+1 2 n+1 2 We shall therefore just state the result corresponding to (1.11), p + q p 2s−1 ζ 1 − s, − ζ 1 − s, 2q 2q q 2Γ (s) πs 2j − 1 2j − 1 (1.16) = − cos − 2π a ζ s, , (2πq)s 2 q 2q j=1 which depends on the relation ∞ x +1 x (−1)n 1 (1.17) 2−sζ s, − q−sζ s, = = L ,s,x . 2 2 (n + x)s 2 n=0 2. Limiting cases We now turn to the limiting case s → n of (1.9) with 1 ≤ n ∈ Z. Wenotethat recently Murty-Saradha [MS] used the Laurent expansion (2.5) at s = 1 of (1.9) to deduce the Gauss formula for the digamma function. To state our main result, we introduce Milnor’s gamma function γn(x) ([Mil]) through ∂ (2.1) γ (x)= ζ (−n, x) . 1+n ∂n License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use DIRICHLET SERIES AT RATIONAL ARGUMENTS 1227 Theorem 2. For q>1,n∈ N and p ∈ Z, we have q−1 −n 2πi a p a q e q ζ n, q a=1 ⎧ ⎫ ⎨ p ⎬ in+1 Bn q p p = πi + γ − (−1)nγ 1 − (2π)1−n(n − 1)! ⎩ n n q n q ⎭ ⎧ ⎧ ⎫ ⎪ n+2 ⎨ p ⎬ ⎪ − 2 Bn ⎪ ( 1) q − p − − p | ⎪ 1−n π i γn γn 1 , 2 n ⎨ (2π) (n − 1)! ⎩ n q q ⎭ (2.2) = ⎧ ⎫ ⎪ ⎪ n+1 ⎨ B p ⎬ ⎪ (−1) 2 n q p p ⎪ − iπ + γn + γn 1 − , 2 |n, ⎩ (2π)1 n (n − 1)! ⎩ n q q ⎭ the case n =1being a singular case to be stated in Corollary 4.
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