The exponential-type generating function of the Riemann zeta-function revisited

Dirichlet series associated with the Poincaré series attached to SL(2,Z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{SL}(2,{{\mathbb {Z}}})$$\end{document} are introduced. Integral representations and transformation formulas are given, which describe the Voronoï-type summation formula for the exponential-type generating function of the Riemann zeta-function. As an application, a new proof of the Fourier series expansion of holomorphic Poincaré series is given.


Introduction
Let s ∈ C. The main object of this paper is the following Dirichlet series: In the present paper, we describe an integral representation of (1.1) along a Hankel contour, which gives a functional relation or transformation formula (Theorems 1.1 and 1.2). As a natural consequence of the functional relation [Theorem 1.2 (1.4)], we obtain a new proof of the Fourier series expansion of the holomorphic Poincaré series attached to SL(2, Z) (Corollary 1.3). For the exponential-type generating function (1.2), our transformation formula [Theorem 1.2 (1.5)] is equivalent to the Voronoïtype summation formula indicated by Katsurada [14,Theorems 3.1 and 3.2].
Historically, the origin of the Dirichlet series (1.1) goes back to the study of Hardy and Littlewood [7]. They proved the functions Segal [20] proved one Voronoï-type summation formula for the integral mean value of Q(x) = ∞ n=1 sin(x/n)n −1 , and discussed the existence of the Bessel series expansion for Q(x) itself. Based on the character analog of the Poisson summation formula due to Berndt [1], Kano [13] derived suitable expressions for ∞ n=1 χ(n) cos(x/n)n −1 and ∞ n=1 χ(n) sin(x/n)n −1 with primitive and non-principal Dirichlet character χ . We note that asymptotic approximations for Q(x) were recently developed by Kuznetsov [15].
In the context of generating functions of the Riemann zeta-function, the Hurwitz zeta-function, the binomial-type generating function for |x| < 1 and s ∈ C \ {1}, was employed by Ramanujan [19] in the generalization of Glaisher's formula [6] Here γ is Euler's constant, ζ(s, x) is the Hurwitz zeta-function, and Γ (s) is the Gamma function.
In recent works [16][17][18], we derived some functional properties of Bessel zetafunctions and a confluent hypergeometric zeta-function. The J -Bessel zeta-function appears in the Fourier series expansion of the Poincaré series attached to SL(2, Z) by the inverse Mellin transform. This fact strongly suggested that our zeta-functions should have a kind of functional equation. The inverse Laplace transform of Weber's first exponential integral was the key ingredient in the proof of the integral expression, which led to the expected transformation formula [16, Theorem 1.1 (1.4)]. Kaczorowski and Perelli treated more general zeta-functions twisted by hypergeometric or Bessel functions, and derived meromorphic continuations of these zeta-functions via the properties of the non-linear twists [9][10][11][12].
Following the techniques employed in [16,17], we drive an integral representation of (1.1) via the inverse Laplace transform of T α e −1/T . The integral expression of ζ exp (s; λ; z) gives a transformation formula when z ∈ R or functional relation when z ∈ C \ R (Theorem 1.2).
Throughout this paper, H denotes the complex upper half-plane. Let J ν (z) be the Bessel function of the first kind, and I ν (z) and K ν (z) be modified Bessel functions (cf. ∞e iθ for an integration taken along a rotated Hankel contour, starting at ∞e i(−2π +θ) , encircling the origin in the positive direction, and returning to the point ∞e iθ . We also denote (c) for an integral over the vertical straight path from c − i∞ to c + i∞. Theorem 1.1 Let λ ∈ C \ {0}, and let z ∈ C with | arg z| < π/2 and 0 < Re(z) ≤ 1.
Let m ∈ Z >0 and k ≥ 4 be an integer. We denote γ (z) and z ∈ H , and use the notation e(z) = exp(2πi z). In a usual manner, we define the m-th Poincaré series attached to SL(2, Z) of weight k by Here the summation is taken over γ = * * c d , a complete system of representation of * *

Corollary 1.3 Fourier series expansion of the holomorphic Poincaré series
Here, the Kloosterman sum is defined as follows:

Corollary 1.4 Under the same assumptions of Theorem 1.1, the integral representation
holds for s ∈ C \ Z, and

Preliminary results
In this section, we establish some Fourier-Mellin integral of K -Bessel function (Lemma 2.2), which are equivalent to the inverse Laplace transforms of T −α exp(−1/T ). We examine this integral transform directly starting from the Mellin-Barnes integrals of the Bessel function. First, we quote Barnes' integral representation of the modified Bessel function of the second kind.
The following Lemma is crucial in the proof of our main results.
Proof Temporarily, we assume | arg T | < π/2, π/2 < θ < π, and π/2 < θ ±arg T < 3π/2. For r > 0, we let Λ r ,θ denote the contour starting on the ray from ∞e −iθ to r e −iθ , encircling the origin with radius r in the positive direction from r e −iθ to r e iθ , and returning on the ray from r e iθ to ∞e iθ . First, we prove the following expression Taking ν = α and Z = 2 √ u in (2.1), we see the right side of (2.3) is equal to The interchange of the order of integration is justified by the absolute convergence of double integrals in (2.4) due to Stirling's formula under the conditions π/2 < θ < π and π/2 < θ ± arg T < 3π/2 for given | arg T | < π/2. After changing variable T u 6 T. Noda to u and shifting back to the integration path to Λ r ,θ , the u-integral in the right side of (2.4) is evaluated by means of (cf. [8,Theorem 8 Hence the right side of (2.4) is equal to for | arg(Z )| < 3π/2, and O-constant depends only on ν. Due to the exponential decay (2.6), the integrand u (α−1)/2 e T u K α−1 (2 √ u) in (2.3) decreases rapidly when |u| → ∞ for every θ such that π/2 < θ + arg T < 3π/2. Therefore, the path of integration (2.3) may be shifted to the rotated Hankel contour as in the integral representation (2.2), which gives the analytic continuation to the domain | arg T | < π with π/2 < θ + arg T < 3π/2. This completes the proof of Lemma 2.2.
In (2.2), it is possible to replace the integrand by using the I ν .
Let r be the small radius around the origin for the path of the above integral. Then the interchange of summation and integration is justified when |Z e u/λ | < 1. Accordingly (3.2) is equal to Here we have used π/sin(π ν) = Γ (ν)Γ (1 − ν) (cf. [4, 1.2 (6)]). The integrals above converge absolutely if Z = e −u/λ for u ∈ U r = u ∈ C | u ∈ (0+) ∞e iθ , hence (3.3) is a holomorphic function of Z and a meromorphic function of s when Z = e −u/λ for u ∈ U r . Therefore, (3.3) gives an analytic continuation of P exp (s; λ; z; Z ) in both Z and s. In particular, P exp (s; λ; z; Z ) is holomorphic at Z = 1. By (2.10), the second integrand in (3.3) is holomorphic except at u = 0, and its residue is .
Proof of Theorem 1.2 In the same manner as the proof of Theorem 1.1 (1.3), we obtain Here ρ, the argument of the Hankel contour, was chosen so as π/2 < ρ + arg{(n + 1 − z)/(−λ)} < 3π/2 for n ∈ Z ≥0 . For z ∈ H with 0 < Re(z) ≤ 1, we may choose ρ as 0 < ρ − arg λ < π/2 in (3.4), and may choose θ as π/2 < θ − arg λ < π in (1.3). Let K(s; λ; z; u) be the integrand in (3.4). Rotating the Hankel contour from ∞e iθ in the positive direction, we obtain Res u=2π e − 3 2 πi nλ K(s; λ; z; u) Applying (2.9) and I ν (ze inπ ) = e inπν I ν (z) for n ∈ Z (cf. [5, 7.11 (44)]), we arrive at the relation formula (1.4). Next, we deform the Hankel contour (1.3) so as to prove the transformation formula (1.5). Let x ∈ R with 0 < x ≤ 1 and λ ∈ R >0 . Temporarily, we assume s < −1. Let R be a sufficiently large integer and C R be the path of integration starting at negative infinity on the real axis, encircling the origin with the radius 2π(R + 1/2)λ in the positive direction, and returning to the starting point. In the transforming the path of integral, the contour C R passes simple poles at u = 2πinλ (n = ±1, ±2, . . . , ±R). Then, by the residue theorem, we have   Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.