Integral Transforms in Number Theory ^†

Abstract

Integral transforms play a fundamental role in science and engineering. Above all, the Fourier transform is the most vital, which has some specifications—Laplace transform, Mellin transform, etc., with their inverse transforms. In this paper, we restrict ourselves to the use of a few versions of the Mellin transform, which are best suited to the treatment of zeta functions as Dirichlet series. In particular, we shall manifest the underlying principle that automorphy (which is a modular relation, an equivalent to the functional equation) is intrinsic to lattice (or Epstein) zeta functions by considering some generalizations of the holomorphic and non-holomorphic Eisenstein series as the Epstein-type Eisenstein series, which have been treated as totally foreign subjects to each other. We restrict to the modular relations with one gamma factor and the resulting integrals reduce to a form of the modified Bessel function. In the H-function hierarchy, what we work with is the second simplest $H_{1,1}^{1,1}\leftrightarrow H_{0,2}^{2,0}$, with H denoting the Fox H-function.

1. Introduction and the Underlying Flow of Ideas

Integral transforms are extremely useful in many fields of science. In areas where the signal is studied, this is expressed as “from time domain $\left\{t\right\}\subset \mathbb{R}$ to frequency domain $\left\{s\right\}\subset \mathbb{C}$, $s=\sigma +j\omega$ and back” ($\omega$ means the frequency and $j=\sqrt{-1}$, which are used mainly in electrical engineering). This is an essential technique to study complicated phenomena in the time domain in the frequency domain, in which phenomena are much easier, and then feed back the results into the time domain. A rather common procedure in many fields of mathematics, where there is a norm or a means to measure distance, zeta functions play a fundamental role. They are usually given in the form of Dirichlet series (additive), and they sometimes possess the Euler product (multiplicative) when the domain in question has a unique factorization property. For treating Dirichlet series, the Mellin transforms are best suited. In many areas of number theory, including applied disciplines, those zeta functions often appear that satisfy the functional equation but do not have Euler products as a lattice zeta function or an Epstein zeta function.

Here we restrict to the use of Mellin transforms in analytic theory of algebraic numbers developed by eminent mathematicians including Riemann, Hecke, Bochner, Hardy, Chowla, Selberg, Weil, et al.

We take up the most famous and basic transforms including the Hecke gamma transform, the Hardy and the beta transform and the Hecke transform. We state a few versions of the Fourier–Bessel expansion—the Chowla–Selberg integral formula, which is one of the most widely used modular relations, equivalent to the functional equation. For a more general case of the Riemann–Hecke–Bochner (RHB) correspondence, which is another name of the modular relation, there is a rather comprehensive book [1] and its sequel [2]. The latter is an exposition of the specified aspects of the former, in which the modular relations are expressed in terms of the Meijer G- and Fox H-functions. Since most of the results are conveniently expressed in G-functions, we refer to the G-function hierarchy. The simplest member in the G-function hierarchy is $G_{0,1}^{1,0}\leftrightarrow G_{0,1}^{1,0}$, which is essentially the automorphy of modular forms. Both [1,2] arrange the contents according to hierarchy and are in a similar spirit to [3] (where the hierarchy depends on the complexity of groups acting on the differential equations). The present paper is a concise and reader-friendly exposition of the simpler part of [2] on the second simplest member of the hierarchy, which we hope will liquidate the burden of grasping the contents ranging to definite binary quadratic forms or imaginary quadratic fields. The G-function considered in the hierarchy of Equation (48) in the paper is up to $G_{0,2}^{2,0}$, or the Bessel functions of the third kind. Higher ones correspond to hypergeometric functions and more involved classes.

Our aim (and hope) is that the paper can be used as a chance for igniting new research by glancing at a collection of loosely relevant formulas. One example came from the relation $\psi \left(z\right)-\psi (1-s)=-\pi cot\pi z$, and another is stated in Remark 1, (i).

To slightly higher number fields including real quadratic fields, cyclotomic fields (save for the case of [4] on a general number field) and the associated Fourier–Whittaker expansion, etc., we hope to return elsewhere. We try to minimize the contents of algebraic number theory, such as class number, the Kronecker limit formula, the Lerch–Chowla–Selberg formula, etc., and emphasize the use of special functions as integral transforms so that we hope the paper will be useful to master the technique of analytic theory of numbers and can go through other works including [5], etc.

For special functions we refer the reader to [6,7,8,9], etc. and for tables, [10,11,12,13], etc.

In Section 2 we use some knowledge of algebraic number theory and modular forms for which we refer to [14,15], respectively.

Throughout the paper, we use the symbol

$$s=\sigma +it,\sigma =\mathrm{Re}s,t=\mathrm{Im}s$$

for the complex variable by the tradition of Riemann. Additionally, the variable $z=x+iy$, etc. is used.

1.1. The Hecke Gamma Transform

Here we shall briefly state some relevant information.

We quote [2] [Proposition 2.0.2], which will explain the background of the Hecke gamma transform. The gamma function is ubiquitously used, and there are several definitions. The definition as the principal solution to the difference equation is often used á la Artin (Bohr–Mollerup theorem). To treat the case of cyclotomic fields, we need Deninger’s R-function, which is the higher derivatives of the Hurwitz zeta function; this is hoped to be treated on another occasion. We introduce the gamma function as the Mellin transform of the rapidly decaying function $e^{-t}$ $\Gamma \left(s\right)={\int }_0^{\infty }{t}^{s-1}{e}^{-t}\mathrm{d}t$, $\sigma >0$, which we restate in the Mellin transform pair

$$\Gamma \left(s\right)={\int }_0^{\infty }{\xi }^s{e}^{-\xi }{\displaystyle \frac{\mathrm{d}\xi }{\xi }},e^{-x}={\displaystyle \frac{1}{2\pi i}}\underset{\left(c\right)}{\int }{x}^{-z}\Gamma \left(z\right)\mathrm{d}z$$

valid for $\sigma >0$ resp. $Rex>0$, $0<c$ extends to the right-half plane (RHP)

$$\mathrm{Re}x\ge 0,x\ne 0:0\ne x=\left|x\right|e^{i\theta },-{\displaystyle \frac{\pi }{2}}\le \theta \le {\displaystyle \frac{\pi }{2}}$$

for $0<\sigma <1$ resp. $0<c<1$ as the Hecke gamma transform and its inversion

Proposition 1.

The Hecke gamma transform

$$x^{-s}\Gamma \left(s\right)={\int }_0^{\infty }{\xi }^s{e}^{-x\xi }{\displaystyle \frac{\mathrm{d}\xi }{\xi }}$$

(1)

which holds for $\mathrm{Re}x>0,\sigma >0$ extends to $\mathrm{Re}x=0,x\ne 0,0<\sigma <1$ and the purely imaginary case $x=iy$, $0<y$ amounts to the one-sided Fourier transform

$$e^{-{\displaystyle \frac{\pi i}{2}}s}{y}^{-s}\Gamma \left(s\right)={\int }_0^{\infty }{\xi }^s{e}^{-iy\xi }{\displaystyle \frac{\mathrm{d}\xi }{\xi }}=\sqrt{2\pi }{F}_{+}\left[{·}^{s-1}\right]\left(y\right).$$

(2)

Or as cosine and sine transforms,

$$\begin{array}{c}\hfill C\left[{·}^{s-1}\right]\left(y\right)=\sqrt{{\displaystyle \frac{2}{\pi }}}{y}^{-s}cos{\displaystyle \frac{\pi }{2}}s\Gamma \left(s\right),S\left[{·}^{s-1}\right]\left(y\right)=\sqrt{{\displaystyle \frac{2}{\pi }}}{y}^{-s}sin{\displaystyle \frac{\pi }{2}}s\Gamma \left(s\right).\end{array}$$

Remark 1.

(i) In literature, the importance of the sine and cosine Mellin transforms has not been noticed. One spectacular example is that the functional equation for the Riemann zeta function is equivalent to the Fourier expansion ${\overline{B}}_2\left(t\right)={\displaystyle \frac{1}{{\pi }^2}}{\sum }_{n=1}^{\infty }{\displaystyle \frac{cos2\pi nt}{n^2}}$, where ${\overline{B}}_2\left(t\right)$ is the periodic extension of the 2nd Bernoulli polynomial. In the proof, the cosine transform plays the vital role [16].

(ii) We extract part of [2] [Remark 2.0.3]. [17] [pp. vii–x] views Equation (2) as the Fourier transform

$${\int }_0^{\infty }{e}^{2\pi i\xi x}{\xi }^{s-1}\mathrm{d}\xi ={(2\pi e^{{\displaystyle \frac{\pi i}{2}}\mathrm{sgn}x})}^{-s}\Gamma \left(s\right){\left|x\right|}^{-s},$$

(3)

which sends the power ${\xi }^{s-1}$ to the power ${\left|x\right|}^{-s}$. This led M. Sato to the search for systematically finding polynomials $f\left(x\right)$ over a vector space V whose complex power is sent to a complex power of another polynomial over the dual space $V^{*}$ under the Fourier transform. Sato found that there is a big-scale action of groups and was led to establishing the theory of prehomogeneous vector spaces (PHVs). This enhanced systematic discovery of zeta functions that satisfy the functional equation. This is a gigantic and influential area equal to and independent of Langlands theory, but there is very little literature in English; cf., e.g., [18]. As with [3], we see that the study of associated group actions will help to reveal the hidden structure.

Let

$$0<{\lambda }_1<{\lambda }_2<\dots \to \infty ,0<{\mu }_1<{\mu }_2<\dots \to \infty$$

be increasing sequences of real numbers. For complex sequences $\left\{a_n\right\},\left\{b_n\right\}$ form the Dirichlet series

$$\phi \left(s\right)=\sum _{n=1}^{\infty }{\displaystyle \frac{a_n}{{\lambda }_n^s}}\mathrm{and}\psi \left(s\right)=\sum _{n=1}^{\infty }{\displaystyle \frac{b_n}{{\mu }_n^s}}$$

(4)

which we assume are absolutely convergent for $\sigma >{\sigma }_a^{*}$ and $\sigma >{\sigma }_b^{*}$, respectively.

We introduce the modular type functions corresponding to the Dirichlet series, Equation (4), for a positive constant A

$$f\left(\tau \right)=\sum _{n=0}^{\infty }{a}_n{e}^{Ain\tau }\mathrm{and}g\left(\tau \right)=\sum _{n=0}^{\infty }{b}_n{e}^{Ain\tau },\tau \in \mathcal{H}$$

which are absolutely convergent and satisfy the (modular) transformation formula

$$f\left(\tau \right)=C{\left({\displaystyle \frac{\tau }{i}}\right)}^{-r}g\left(-{\displaystyle \frac{1}{\tau }}\right).$$

(5)

Here,

$$\mathcal{H}=\left\{\tau =x+iy|y>0\right\}=\left\{z=x+iy|y>0\right\}$$

indicates the upper half-plane. Hereafter we use the symbols $\tau$ and z interchangeably, indicating $x+iy$ ($y>0$).

Lemma 1

(Hecke). That the Dirichlet series (Equation (4)) satisfy the condition that $A^{-s}\Gamma \left(s\right)\phi \left(s\right)+{\displaystyle \frac{a_0}{s}}+{\displaystyle \frac{C{b}_0}{r-s}}$ is BEV (bounded in every vertical strip) and satisfies the functional equation

$$A^{-s}\Gamma \left(s\right)\phi \left(s\right)=C{A}^{-(r-s)}\Gamma (r-s)\psi (r-s).$$

(6)

is equivalent to Equation (5). We refer to (6) as the Hecke (type) functional equation (HTFE).

Lemma 1 is a slight generalization of [19] [Theorem 1, p. I-5], which is a handy statement of Hecke’s epoch-making discovery [20,21]. In most cases $C=1$.

Compare Ogg’s statement in [19] [p. I-7] with Weil’s interpretation [22] that this is a revival of the theory of automorphic forms.

As Hecke’s Lemma 1 is the upper half-plane version, the Bochner relation

$$G_{0,1}^{1,0}\leftrightarrow G_{0,1}^{1,0}$$

is the right half-plane version of the modular relation. One of the most famous examples of the Bochner modular relation is Ramanujan’s formula, cf. [23].

Ref. [2] [Exercise 2.0.5] gives a proof of the functional equation

$${\pi }^{-{\displaystyle \frac{s}{2}}}\Gamma \left({\displaystyle \frac{s}{2}}\right)\zeta \left(s\right)={\pi }^{-{\displaystyle \frac{1-s}{2}}}\Gamma \left({\displaystyle \frac{1-s}{2}}\right)\zeta (1-s)$$

(7)

for $\sigma <0$. We outline Riemann’s first proof depending on the theta-transformation formula

$$e^{\pi i\tau x^2}\vartheta (\tau ,\tau x)=\sqrt{{\displaystyle \frac{i}{\tau }}}\vartheta \left(-{\displaystyle \frac{1}{\tau }},x\right),$$

(8)

where

$$\vartheta (\tau ,x)={\vartheta }_3(\tau ,\pi x)=\sum _{n=-\infty }^{\infty }{e}^{2\pi inx}{q}^{n^2},q=e^{\pi i\tau }$$

(9)

and ${\vartheta }_3$ is one of 4 theta series of Jacobi, cf., e.g., [24,25,26].

Let

$$\omega \left(t\right)={\displaystyle \frac{1}{2}}\vartheta (it,0)-{\displaystyle \frac{1}{2}}=\sum _{n=1}^{\infty }{e}^{-\pi n^{2}t},\mathrm{Re}t>0.$$

Then, Equation (8) reads for $\omega \left(t\right)$

$$\omega \left(t^{-1}\right)=-{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}\sqrt{t}+\sqrt{t}\omega \left(t\right).$$

(10)

By the Hecke gamma transform, we obtain

$${\pi }^{-{\displaystyle \frac{s}{2}}}\Gamma \left({\displaystyle \frac{s}{2}}\right)\zeta \left(s\right)={\int }_0^{\infty }{t}^{{\displaystyle \frac{s}{2}}}\omega \left(t\right){\displaystyle \frac{\mathrm{d}t}{t}}.$$

(11)

Dividing the integral in Equation (11) into two parts, ${\int }_0^1$ and ${\int }_1^{\infty }$, making the change in variable $t\leftrightarrow {\displaystyle \frac{1}{t}}$ and applying we deduce Equation (7).

1.2. Beta and Hardy Transform and Fourier–Bessel Expansion

The following lemma is a basis of the beta transform also called the beta integral. In previous studies it used to be called the Mellin–Barnes integral, but this refers to a much wider class of functions, cf. [2] [Remark 2.4.1] and the passage above Theorem 2. Results on the beta transform are culminated in [27] and are a basis of the Fourier–Bessel expansion, Equation (26), cf. [1]. It is stated in Theorem 2 to the effect that the beta transform and the Hardy transform lead to the same Fourier–Bessel expansion, Equation (26). In view of this, we may claim that the beta transform was first used by Hardy [28] in the form of the Hardy transform. In the form of the beta integral, this was used by Koshlakov [29] and extensively used by Berndt [30].

Lemma 2

([2] Lemma 2.4.1). The beta transform reads

$${(1+x)}^{-s}={\displaystyle \frac{1}{2\pi i}}{\int }_{\left(c\right)}{\displaystyle \frac{\Gamma (z+s)\Gamma (-z)}{\Gamma \left(s\right)}}{x}^z\mathrm{d}z={\displaystyle \frac{1}{\Gamma \left(s\right)}}{G}_{1,1}^{1,1}\left(x^{-1}\left|\begin{array}{c}1\\ s\end{array}\right.\right)$$

(12)

for $x>0$, $-\mathrm{Re}s<c<0$, where $\left(c\right)$ signifies the vertical Bromwich path $z=c+it,$$-\infty <t<\infty$.

Cf. Remark 3.

The following notation will be used in connection with a positive definite binary quadratic form Q:

$$Q=Q(m,n)=a{m}^2+bmn+c{n}^2=a{|m+n\tau |}^2,\tau =x+iy={\displaystyle \frac{b+i\sqrt{|\Delta |}}{2a}},$$

(13)

where

$$a={\displaystyle \frac{\sqrt{|\Delta |}}{2\mathrm{Im}\tau }},\Delta =b^2-4ac<0,$$

(14)

so that

$$c={a\left|\tau \right|}^2,b=2a\mathrm{Re}\tau ,x={\displaystyle \frac{b}{2a}}.y={\displaystyle \frac{\sqrt{|\Delta |}}{2a}}.$$

(15)

Then in conformity with Equation (13), $\tau$ is the solution of the quadratic equation with imaginary part positive,

$$a{\tau }^2-b\tau +c=0.$$

(16)

Then, the Epstein zeta function (EZF) $Z(s,Q)$ associated to a positive definite quadratic form Q is defined by

$$\begin{array}{c}\hfill Z(s,Q)={\sum _{m,n}}^{\prime }{\displaystyle \frac{1}{Q{(m,n)}^s}}=a^{-s}{\zeta }_{{\mathbb{Z}}^2}(s,\tau ),\end{array}$$

(17)

where the prime on the summation sign means the omission of the case $m=n=0$ and the Epstein-type Eisenstein series (EES) is defined by

$${\zeta }_{{\mathbb{Z}}^2}(s,\tau )={\sum _{m,n}}^{\prime }{\displaystyle \frac{1}{{|m+n\tau |}^{2s}}}={\sum _{m,n}}^{\prime }{\displaystyle \frac{1}{{\left({(m+xn)}^2+{\left(yn\right)}^2\right)}^s}},$$

(18)

where $\sigma >1,\tau \in \mathcal{H}$ and the prime on the summation sign means that $m=n=0$ is excluded, and where $\mathcal{H}$ is the upper half-plane. Cf. e.g., [31,32], etc. We usually refer to both of them as the equality

$$\begin{array}{c}\hfill Z(s,Q)=a^{-s}{\zeta }_{{\mathbb{Z}}^2}(s,\tau ).\end{array}$$

Theorem 1

([2] Theorem 2.4.1). Under the notation of Equations (13)–(18), in particular, $\tau =x+iy$, $y={\displaystyle \frac{\sqrt{|\Delta |}}{2a}}$, we have the Chowla–Selberg integral formula

$$\begin{array}{cc}\hfill & \Gamma \left(s\right)Z(s,Q)=\Gamma \left(s\right)a^{-s}{\zeta }_{{\mathbb{Z}}^2}(s,\tau )\hfill \\ \hfill & =2\Gamma \left(s\right)\zeta \left(2s\right)a^{-s}+{\displaystyle \frac{2^{2s}{a}^{s-1}\sqrt{\pi }}{{|\Delta |}^{s-\frac{1}{2}}}}\Gamma \left(s-{\displaystyle \frac{1}{2}}\right)\zeta (2s-1)\hfill \\ \hfill & {\displaystyle \frac{8{\pi }^s{2}^{s-\frac{1}{2}}}{\sqrt{a}{|\Delta |}^{\frac{s}{2}-\frac{1}{4}}}}\sum _{n=1}^{\infty }{n}^{s-{\displaystyle \frac{1}{2}}}{\sigma }_{1-2s}\left(n\right)cos2\pi nx{K}_{s-{\displaystyle \frac{1}{2}}}\left(2\pi yn\right),\hfill \end{array}$$

(19)

where

$${\sigma }_{\alpha }\left(n\right)=\sum _{d|n}{d}^{\alpha }$$

is the sum-of-divisors function and $K_{\nu }$ is the K-Bessel function Equation (22), cf. [2] [Section 3.1]. Equation (19) is equivalent to the functional equation

$${\left({\displaystyle \frac{2\pi }{\sqrt{|\Delta |}}}\right)}^{-s}\Gamma \left(s\right)Z\left(s\right)={\left({\displaystyle \frac{2\pi }{\sqrt{|\Delta |}}}\right)}^{-(1-s)}\Gamma (1-s)Z(1-s).$$

(20)

Hardy’s formula [28] (or Hardy’s integral) reads

$${\psi }_a\left(z\right)=z^{{\displaystyle \frac{a+1}{2}}}{K}_{{\displaystyle \frac{a+1}{2}}}\left(z\right)={\int }_0^{\infty }{e}^{-t^2-{\displaystyle \frac{z^2}{t^2}}}{t}^{a+1}{\displaystyle \frac{\mathrm{d}t}{t}},\mathrm{Re}z>0.$$

(21)

and we call an operation involving this the Hardy transform. As shown in Comments on the paper in Hardy’s Collected Papers, this paper was overlooked by Watson in his treatise [8] although it is essentially [8] [(15), p. 183]:

$$K_{\nu }\left(z\right)={\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{z}{2}}\right)}^{\nu }{\int }_0^{\infty }{e}^{-t-{\displaystyle \frac{z^2}{4t}}}{t}^{-\nu }{\displaystyle \frac{\mathrm{d}t}{t}},\mathrm{Re}\nu >-{\displaystyle \frac{1}{2}},|argz|<{\displaystyle \frac{\pi }{4}}.$$

(22)

By the change in variable, we arrive at a handy formula which leads to Hardy’s Equation (21)

$$\begin{array}{c}\hfill {\int }_0^{\infty }{t}^s{e}^{-\alpha t^p-{\displaystyle \frac{\beta }{t^p}}}{\displaystyle \frac{\mathrm{d}t}{t}}={\displaystyle \frac{2}{p}}{\left({\displaystyle \frac{\beta }{\alpha }}\right)}^{{\displaystyle \frac{s}{2p}}}{K}_{{\displaystyle \frac{s}{p}}}\left(2\sqrt{\alpha \beta }\right),\mathrm{Re}\alpha >0,\mathrm{Re}\beta >0,0\ne p\in \mathbb{R}.\end{array}$$

(23)

The special case of Equation (23) with $p=1$ can be found on [12] [p. 85].

The symmetry property

$$K_{-\nu }\left(z\right)=K_{\nu }\left(z\right)$$

(24)

and the reduction formula

$$K_{\pm {\displaystyle \frac{1}{2}}}\left(z\right)=\sqrt{{\displaystyle \frac{\pi }{2z}}}{e}^{-z}.$$

are quite often used. Indeed, the functional Equation (20) follows from Equation (9), (24) and the property of the sum-of-divisors function.

Bellman defines the Voronoĭ function (or the generalized Bessel function) [24] [(1.1)]

$$V_a(x,y)={\int }_0^{\infty }{e}^{-\pi x^{2}t-{\displaystyle \frac{\pi y^2}{t}}}{t}^{-a}\mathrm{d}t=2{\left({\displaystyle \frac{y}{x}}\right)}^{1-a}{K}_{1-a}\left(2\pi xy\right),$$

(25)

which reduces for $\mathrm{Re}x>0$ to [33] [(3)]

$$W_a\left(x\right)=V_a\left(\sqrt{{\displaystyle \frac{x}{\pi }}},{\displaystyle \frac{1}{\sqrt{\pi }}}\right)={\int }_0^{\infty }{e}^{-xt-{\displaystyle \frac{1}{t}}}{t}^{-a}\mathrm{d}t=2x^{{\displaystyle \frac{a-1}{2}}}{K}_{a-1}\left(2\sqrt{x}\right).$$

Bellman [33] seems to be the first who explicitly mentioned Hardy’s paper [28]. But he emphasized the Voronoĭ function and used a special case of Hardy’s formula.

Prior to this, Chowla and Selberg [34] [p. 371, (4)] used the Hardy transform without mentioning Hardy’s integral. In Part 2 of their paper [35] [(71) and (72)], the use of the Hardy transform is evident.

The study of Epstein zeta functions used to be for determining the behavior of the class number $h(-\Delta )$ of imaginary quadratic fields (or positive definite binary quadratic forms). Chowla and Selberg [34] mention Deuring’s paper [36] and that the formula led Heilbronn [37] to prove a conjecture on $h(-\Delta )$. This led to Siegel’s famous and important result [38]. Cf. e.g., [39].

The terminology “beta transform” in Lemma 2 arises in contrast to the Hecke gamma transform (HGT), Equation (1).

Koshlyakov used the beta transform extensively in [29] which may be described as the theory of special functions related to the K-Bessel function.

Independently of Koshlyakov, Berndt [40,41] resp. [30] made an extensive study of perturbed Dirichlet series, Equation (27), and the beta transform resp. the Hardy transform, deriving the Fourier–Bessel Equation (26). Ref. [41] [Theorem 3.1] establishes the equivalence of the Hecke functional equation and the Fourier–Bessel expansion for the case of the general gamma factor.

The modular relation that is the most useful and relevant to us is the Fourier–Bessel expansion

$$\begin{array}{cc}{A}^{-s}\Gamma \left(s\right)\phi (s,\alpha )=\hfill & 2{\alpha }^{{\displaystyle \frac{r-s}{2}}}\sum _{n=1}^{\infty }{b}_n{\mu }_n^{-{\displaystyle \frac{r-s}{2}}}{K}_{s-r}\left(2A\sqrt{\alpha {\mu }_n}\right)\hfill \\ \hfill & +A^{-s}{\int }_0^{\infty }{e}^{-\alpha u}{u}^{s-1}\mathrm{P}\left({\displaystyle \frac{u}{A}}\right)du\hfill \end{array}$$

(26)

with $\alpha >0,\sigma >max\{r-{\displaystyle \frac{1}{2}},-1\},s\ne 0$, where

$$\phi (s,\alpha )=\sum _{n=1}^{\infty }{\displaystyle \frac{a_n}{{({\lambda }_n+\alpha )}^s}}$$

(27)

denotes the (Hurwitz type) perturbed Dirichlet series associated to $\phi$.

To study the shifted sequence ${\lambda }_n+\alpha$ as an auxiliary means has turned out to be the most effective since it has an effect of amplifying the features of the original sequence. This includes the PNT for an arithmetic progression, lattice zeta functions, class field theory, etc., Cf. [2] [Section 5.3].

Remark 2.

The Chowla–Selberg integral formula (CSIF) in Theorem 1 is the Fourier–Bessel expansion. To memorize the importance of their result, we gave it the special term (CSIF) and will refer to this term to mean the Fourier–Bessel expansion. Ref. [2] [Chapter 2] contains rich material on this subject including the Kronecker limit formula, the Lerch–Chowla–Selberg formula, and the Fourier–Whittaker expansion for non-holomorphic automorphic forms (for which there is no Fourier-expansion). This is visible in the G-function hierarchy, Equation (48), below.

Ref. [42] develops the theory of Epstein zeta functions by the beta transform method. For the case of Eisenstein series, cf. [31,34,35], etc.

Then the Hardy transform is used in [43] to deal with the Epstein zeta function. The remark that Hardy [28] seems to be the first who used the beta transform (in treating lattice sums) is to be interpreted to mean the Hardy transform as well as the beta transform lead to the Fourier–Bessel expansion (Theorem 2).

The method was also adopted by Katsurada and Matsumoto under the name of the Mellin-Barnes integrals in the context of applications of Atkinson’s dissection [44,45]. But this terminology usually refers to more general integrals as synonymously as the Fox H-function [10] [I, p. 39] and [6].

Remark 3.

We shall elucidate the reason why the beta transform, Lemma 2 has been so much used in treating shifted sequences, which we denote $1+x$ symbolically.

The key lies in the merit of the gamma transform Equation (1). It puts x and $1+x$ in the denominator to the numerator in the exponential function, respectively. In the case of $1+x$, the merit is more visible; it is additive, and the exponential function decomposes. If we deal with a sum, then we often need to adopt dissection of the sum so as to separate the 1-part and $x\ne 0$-part. This is the origin of the dissections. In the beta transform, it can separate 1 and x in the inverse Mellin transform, as can be seen

$$\begin{array}{cc}\hfill & {\int }_0^{\infty }{\xi }^s{e}^{-(1+x)\xi }{\displaystyle \frac{\mathrm{d}\xi }{\xi }}\hfill \\ \hfill & =\Gamma \left(s\right){(1+x)}^{-s}\hfill \\ \hfill & ={\displaystyle \frac{1}{2\pi i}}{\int }_{\left(c\right)}\Gamma (z+s)\Gamma (-z)x^z\mathrm{d}z=G_{1,1}^{1,1}\left(x^{-1}\left|\begin{array}{c}1\\ s\end{array}\right.\right).\hfill \end{array}$$

In the light of the governing principle $G_{1,1}^{1,1}\leftrightarrow G_{0,2}^{2,0}$, we may say that in the beta transform, one needs to apply the functional equation to transform the integral into $G_{0,2}^{2,0}$. Cf. the proof of [2] [Theorem 2.4.1].

On the other hand, in the Hardy–Hecke transform, one starts from the gamma transform, the Mellin transform, and incorporates a modular relation to arrive at the Hecke integral—the final form. Cf. Section 1.3 and the proof of Theorem 3.

We summarize the contents of the above remark in the following theorem and describe it symbolically as “The Hardy and beta transforms are reverse processes”. For notation etc., cf. [2] [Section 3.1].

Theorem 2

([2] Theorem 3.1.1). The Hardy transform, resp. the beta transform, is based on the Mellin transform, resp. the inverse Mellin transform, leading to the Fourier–Bessel expansion, Equation (26). More precisely, if one applies the beta transform to the perturbed Dirichlet series and applies the modular relation to the x-part, then one arrives at Equation (26). On the other hand, if one applies the gamma transform to $\Gamma \left(s\right)\phi \left(s\right)$ and applies a certain transformation formula to the x-part, then one arrives at the Hardy integral, leading to the K-Bessel function expression.

That the Hardy transform and the beta transform are reverse processes leading to the Fourier–Bessel expansion was first remarked by Berndt [41] [Theorem 8.1, p. 342]. Cf. [46]. Cf. [2] [3.1] and Refs. [4,44,45,47,48,49,50]. Cf. [2] [Table 3.4] and Refs. [25,31,34,35,36,42,51,52,53,54,55,56].

1.3. On Proofs of Theorem 1

We quote a portion of [2] [Table 3.3] (

Table 1. Epstein zeta functions and non-holom. Eisenstein ser.

Zeta	Symbol	A
class Dedekind zeta	${\zeta }_K(s,\mathfrak{C})={\sum }_{\mathfrak{C}\ni \mathfrak{a}\ne 0}{\displaystyle \frac{1}{{\left(N\mathfrak{a}\right)}^s}}$	${\displaystyle \frac{2\pi }{\sqrt{|\Delta |}}}$
Epstein zeta	$Z(s,Q)={\sum }_{m,n}^{\prime }{\displaystyle \frac{1}{{Q(m,n)}^s}}$	${\displaystyle \frac{2\pi }{\sqrt{|\Delta |}}}$
Epstein-type Eisenstein	${\zeta }_{{\mathbb{Z}}^2}(s,\tau )$	${\displaystyle \frac{\pi }{y}}$
Eisenstein–Maass Equation (28)	$E(s,z)=E_0(\tau ,s)={\displaystyle \frac{y^s}{\zeta \left(2s\right)}}{\zeta }_{{\mathbb{Z}}^2}(s,z)$	${\displaystyle \frac{\pi }{y}}$
real-analytic Eisenstein	$E^{*}(s,z)={\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{\pi }{y}}\right)}^{-s}\Gamma \left(s\right){\zeta }_{{\mathbb{Z}}^2}(s,z)$	1

):

Here A is the constant in the Hecke (type) functional Equation (6) and $\mathfrak{C}$ is an absolute ideal class of an imaginary quadratic field K (which has been one of the main objects of research).

Anticipating their roles in Section 2, we introduce Eisestein–Maass series

$$E(s,z)=\sum _{\gamma \in {\Gamma }_1\setminus \Gamma }{y}^s\left(\gamma z\right),y\left(z\right)=\mathrm{Im}z,$$

(28)

where ${\Gamma }_1$ is the stabilizer of ∞. $E(s,z)$ was first introduced by Maass [57] [(106), p. 162] as the Eisenstein series of level N for $\Gamma =\Gamma \left(1\right)$, the full modular group. It follows that

$$\begin{array}{c}\hfill E(s,z)=y^s\underset{\begin{array}{c}c,d=-\infty \\ (c,d)=1\end{array}}{\overset{\infty }{{\sum }^{\prime }}}{|cz+d|}^{-2s}={\displaystyle \frac{y^s}{\zeta \left(2s\right)}}{\zeta }_{{\mathbb{Z}}^2}(s,z).\end{array}$$

(29)

Associated with this is the real-analytic Eisenstein series defined by ([2] [(3.6.1)])

$$\begin{array}{cc}\hfill E^{**}(s,z)& ={\displaystyle \frac{\Gamma \left(s\right)}{2{\pi }^s}}\sum _{{\displaystyle \genfrac{}{}{0pt}{}{m,n\in \mathbb{Z}}{(m,n)\ne (0,0)}}}\frac{y^s}{{|m+nz|}^{2s}}={\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{\pi }{y}}\right)}^{-s}\Gamma \left(s\right){\zeta }_{{\mathbb{Z}}^2}(s,z)\hfill \\ \hfill & ={\pi }^{-s}\Gamma \left(s\right)\mathcal{E}(s,z),\hfill \end{array}$$

(30)

where

$$\mathcal{E}(s,z)={\displaystyle \frac{1}{2}}\zeta \left(2s\right)E(s,z)={\displaystyle \frac{1}{2}}{y}^s{\zeta }_{{\mathbb{Z}}^2}(s,z),$$

(31)

which will appear in Corollary 1.

In the proof [2] [pp. 75–78] the argument is as follows: Collecting the terms with $n=0$ gives $2{\sum }_{m=1}^{\infty }{m}^{-2s}=2\zeta \left(2s\right)$. In the remaining sum

$$S_2:=\sum _{n\ne 0}\sum _{m=-\infty }^{\infty }{\{{(m+nx)}^2+{\left(ny\right)}^2\}}^{-s},$$

(32)

we transform the summand into ${\left(ny\right)}^{-2s}{\left(1+{\left({\displaystyle \frac{m+nx}{ny}}\right)}^2\right)}^{-s}$ and apply the beta integral and at the final stage the beta transform in Lemma 2.

Here we give sketches of two different proofs depending on the Hecke gamma transform Equation (1) ([35]) and the Poisson summation formula ([32]), respectively. Both proofs appeal to the closed form for the resulting integrals, the Hardy integral (33) for the former and the Basset integral (34) for the latter. These proofs are unified in the proof of [2] [Theorem 3.6.2.] in the following description.

Assume as in [2] [Theorem 3.6.2.] asymptotic formulas, the limit conditions and a Fourier transform formula

$${\int }_{-\infty }^{\infty }{e}^{-\pi \alpha x^2}e\left(rx\right)dx=\sqrt{{\displaystyle \frac{1}{\alpha }}}exp\left(-{\displaystyle \frac{\pi }{\alpha }}{r}^2\right),\alpha >0.$$

Then the integral representation (Hecke integral), cf. Equation (23),

$$K_{\nu }\left(z\right)={\displaystyle \frac{1}{2}}{\int }_0^{\infty }exp\left(-{\displaystyle \frac{z}{2}}\left(t+{\displaystyle \frac{1}{t}}\right)\right)t^{\nu }{\displaystyle \frac{\mathrm{d}t}{t}},\mathrm{Re}\nu >-{\displaystyle \frac{1}{2}},|argz|<{\displaystyle \frac{\pi }{4}}$$

(33)

and the formula of Basset

$$\begin{array}{cc}{K}_{\nu }\left(z\right)={\displaystyle \frac{1}{\sqrt{\pi }}}{\left(2z\right)}^s\Gamma \left(\nu +{\displaystyle \frac{1}{2}}\right)& {\int }_0^{\infty }{\displaystyle \frac{cost}{{(t^2+z^2)}^{2\nu +1}}}dt,\\ \mathrm{Re}\nu \ge -{\displaystyle \frac{1}{2}},\left|argz\right|<{\displaystyle \frac{\pi }{2}},\end{array}$$

(34)

follow, and at the same time we obtain the Chowla–Selberg integral formula, Theorem 1, (i). Proof is by giving a closed form for the Fourier coefficients similar to the second proof of Chowla and Selberg [35].

Hereafter we assume above formulas and prove the following formula, which is a combination of [2] [(2.4.30), (3.6.27)]:

$$\begin{array}{cc}\Delta \left(s\right){\zeta }_{{\mathbb{Z}}^2}(s,z)\hfill & =2\zeta \left(2s\right)\Delta \left(s\right)+{\displaystyle \frac{2\sqrt{\pi }\Gamma (s-\frac{1}{2})}{\Gamma \left(s\right)}}\zeta (2s-1)y^{1-2s}\Delta \left(s\right)\hfill \\ \hfill & +{\displaystyle \frac{8{\pi }^s{y}^{\frac{1}{2}-s}}{\Gamma \left(s\right)}}\Delta \left(s\right)\sum _{n=1}^{\infty }{n}^{s-{\displaystyle \frac{1}{2}}}{\sigma }_{1-2s}\left(n\right)cos2\pi nx{K}_{s-{\displaystyle \frac{1}{2}}}\left(2\pi ny\right).\hfill \end{array}$$

(35)

Here we mean $z=x+iy\in \mathcal{H}$. We give a proof of the case $\Delta \left(s\right)=1$, which is [2] [(3.6.27)]. The special case $\Delta \left(s\right)=a^{-s}$ is Theorem 1 and $\Delta \left(s\right)={\left({\displaystyle \frac{y}{\pi }}\right)}^s$ is a variant of Equation (62).

Proof of Theorem 1

(cf. [32]). Here the key is [32] [(2.9)] with $n=1$

$$\begin{array}{cc}\tilde{F}(s,z)\hfill & ={\displaystyle \frac{\sqrt{\pi }\Gamma (s-\frac{1}{2})}{\Gamma \left(s\right)}}{y}^{1-2s}\hfill \\ \hfill & +{\displaystyle \frac{4\sqrt{\pi }{y}^{1-2s}}{\Gamma \left(s\right)}}\sum _{m=1}^{\infty }{\left(\pi my\right)}^{s-{\displaystyle \frac{1}{2}}}cos2\pi mx{K}_{s-{\displaystyle \frac{1}{2}}}\left(2\pi my\right)\hfill \end{array}$$

(36)

for the lattice zeta function ([2] [(2.0.81),(2.4.31)])

$$\tilde{F}(s,z):=\sum _{m=-\infty }^{\infty }{\displaystyle \frac{1}{{|m+z|}^{2s}}}.$$

(37)

Equation (36) is proved by the Poisson summation formula applied to the periodic function $f\left(t\right)={\displaystyle \frac{1}{{|t+z|}^{2s}}}$. Substituting Equation (36) in the dissection

$$\begin{array}{c}\hfill \Delta \left(s\right){\zeta }_{{\mathbb{Z}}^2}(s,z)=2\zeta \left(2s\right)\Delta \left(s\right)+2\Delta \left(s\right)\sum _{n=1}^{\infty }\tilde{F}(s,nz)\end{array}$$

completes the proof. □

Proof of Theorem 1

(cf. [35]). We apply the Hecke gamma transform to each summand of $S_2$ in Equation (32) to obtain

$$S_2={\displaystyle \frac{{\pi }^s}{\Gamma \left(s\right)}}{\int }_0^{\infty }{t}^s\sum _{n\ne 0}{e}^{-\pi {\left(ny\right)}^{2}t}\sum _{m=-\infty }^{\infty }{e}^{-\pi {(m+nx)}^{2}t}{\displaystyle \frac{\mathrm{d}t}{t}}.$$

(38)

We apply a variant of the theta transformation formula to the inner sum. The situation is described in Ogg’s result in [2] [Lemma 2.0.1] and we quote some results. The theta transformation Equation (8) may be expressed as

$$f_j(\tau ,\xi )=\sqrt{{\displaystyle \frac{i}{\tau }}}{f}_k\left(-{\displaystyle \frac{1}{\tau }},\xi \right),\{j,k\}=\{1,2\},$$

(39)

on putting

$$f_1(\tau ,\xi )=e^{\pi i\tau {\xi }^2}\vartheta (\tau ,\tau \xi ),f_2(\tau ,\xi )=\vartheta \left(\tau ,\xi \right).$$

Substituting

$$\begin{array}{cc}\hfill & f_1(it,\xi )=e^{-\pi t{\xi }^2}\vartheta (it,it\xi )=\sum _{m=-\infty }^{\infty }{e}^{-\pi t{(m+\xi )}^2},\hfill \\ \hfill & f_2(it,\xi )=\vartheta (it,\xi )=\sum _{m=-\infty }^{\infty }{e}^{2\pi im\xi -\pi t{m}^2}\hfill \end{array}$$

(40)

in Equation (39), we rewrite the special case of Equation (39) as

$$\begin{array}{cc}\hfill \sum _{m=-\infty }^{\infty }{e}^{-\pi t{(m+\xi )}^2}& ={\displaystyle \frac{1}{\sqrt{t}}}\sum _{m=-\infty }^{\infty }{e}^{2\pi im\xi -{\displaystyle \frac{\pi m^2}{t}}}\hfill \\ \hfill & ={\displaystyle \frac{1}{\sqrt{t}}}\left(1+2\sum _{m=1}^{\infty }cos\left(2\pi m\xi \right)e^{-{\displaystyle \frac{\pi m^2}{t}}}\right)\hfill \end{array}$$

Substituting this in the inner sum in Equation (38), we deduce that

$$\begin{array}{cc}{S}_2\hfill & ={\displaystyle \frac{2{\pi }^s}{\Gamma \left(s\right)}}{\int }_0^{\infty }{t}^{s-{\displaystyle \frac{1}{2}}}\sum _{n=1}^{\infty }{e}^{-\pi {\left(ny\right)}^{2}t}{\displaystyle \frac{\mathrm{d}t}{t}}\hfill \\ \hfill & +{\displaystyle \frac{4{\pi }^s}{\Gamma \left(s\right)}}{\int }_0^{\infty }{t}^{s-{\displaystyle \frac{1}{2}}}\sum _{m,n=1}^{\infty }{e}^{-\pi {\left(ny\right)}^{2}t-{\displaystyle \frac{\pi m^2}{t}}}cos\left(2\pi mnx\right){\displaystyle \frac{\mathrm{d}t}{t}},\hfill \end{array}$$

(41)

which is [35] [(9)]. This reduces to

$$\begin{array}{cc}\hfill & {\pi }^{-s}\Gamma \left(s\right)S_2\hfill \\ \hfill & =2\Gamma \left(s-{\displaystyle \frac{1}{2}}\right)\zeta (2s-1){\left(y\right)}^{1-2s}+4\sum _{m,n=1}^{\infty }cos\left(2\pi mnx\right)H_{m,n}\left(s\right),\hfill \end{array}$$

where

$$H_{m,n}\left(s\right)={\int }_0^{\infty }{t}^{s-{\displaystyle \frac{1}{2}}}{e}^{-\pi {\left(ny\right)}^{2}t-{\displaystyle \frac{\pi m^2}{t}}}{\displaystyle \frac{\mathrm{d}t}{t}}$$

is the Hecke integral (33) and Equation (46). Hence it is equal to $2{\left({\displaystyle \frac{ny}{m}}\right)}^{s-{\displaystyle \frac{1}{2}}}{K}_{s-{\displaystyle \frac{1}{2}}}\left(2\pi mny\right)$. This completes the proof of Equation (35) (with $\Delta \left(s\right)=a^{-s}$). □

Remark 4.

Lewittes [58] considered the lattice zeta function

$$F(s,z)=\sum _{m=-\infty }^{\infty }{\displaystyle \frac{1}{{(m+z)}^s}},$$

(42)

which is absolutely and uniformly convergent for $\sigma >1$ ($\mathrm{Im}z\ne 0$) and the argument is restricted to

$$-\pi \le argz<\pi ,0\ne z\in \mathbb{C},-1=e^{-\pi i}.$$

Using the Lipschitz summation formula, Lewittes proved a Lambert series expression

$$F(s,\tau )={\displaystyle \frac{{(-2\pi i)}^s}{\Gamma \left(s\right)}}\sum _{n=1}^{\infty }{n}^{s-1}{e}^{2\pi i\tau n}.$$

(43)

It should be noted that in the above two proofs, the Poisson summation formula and the theta transformation formula [59] are used, respectively, and that in the proof of (43), the Lipschitz summation formula is used. These three are more or less equivalent to the functional equation for the Riemann zeta function, cf. [59].

$F(2s,z)$ in Equation (42) is closely related to Equation (37) in the context of the Kronecker limit formula and the resulting (analog of) Dedekind eta-function, cf. Section 2. Equation (37) leads to the $log\left|\eta \right(z\left)\right|$ while Equation (42) leads to the zeta-regularization of the determinant associated with the Fermionic strings (based on the philosophy of H. M. Stark [60] that the value at $s=1$ of a zeta function may be obtained from that of the derivative at $s=0$—under implicit assumption of the functional equation). Cf. [61].

Towards the end of Section 1.1, one of two of Riemann’s proofs of the functional equation is given. There, there is a need to divide the range of the integral into two so that there appear t and $t^{-1}$, which determine the group structure of the positive reals $t\leftrightarrow t^{-1}$. The Hardy integral contains both t and $t^{-1}$ and therefore contains a bud of the modular relation.

1.4. Hecke Transform

We state Hecke’s generalization of the Hardy transform unifying Equations (22) and (45) as is expounded in [62] [pp. 55–58]. This was first introduced in [63] [p. 114, (12) and (13)]. For $\alpha ,\beta \in \mathbb{C}$ and $a,b>0$, we have

$$\begin{array}{c}\hfill {\int }_0^{\infty }{t}^{\beta -\alpha }{e}^{-{\displaystyle \frac{a}{t}}-bt}{\displaystyle \frac{\mathrm{d}t}{t}}={\displaystyle \frac{1}{2\pi i}}{\int }_{\left(c\right)}\Gamma \left(s+\alpha \right)\Gamma \left(s+\beta \right)a^{-s-\alpha }{b}^{-s-\beta }\mathrm{d}s,\end{array}$$

(44)

where $\left(c\right)$ separates the poles of the integrand (in Hecke’ paper, $c>-\mathrm{Re}\alpha ,-\mathrm{Re}\beta$). A prototype of Equation (44) with $b=1$ can be found in [64] [p. 325, (65)].

Hecke has made important applications of Equation (44) in the theory of real quadratic fields. Siegel [65] [(5)] may be one of the first who applied the (special case of) Hecke transform. It is further generalized by Rademacher [66] [p. 352, Hilfssatz 13] to algebraic number fields of higher degree and played an important role.

As the Mellin inverse we have the inverse Heaviside integral which most often appears in the context of Fourier–Bessel expansion, [10] [I, p. 316, (4)]:

$$\begin{array}{cc}2K_{\nu }\left(2x\right)\hfill & ={\displaystyle \frac{1}{2\pi i}}{\int }_{\left(c\right)}\Gamma \left(s+{\displaystyle \frac{\nu }{2}}\right)\Gamma \left(s-{\displaystyle \frac{\nu }{2}}\right)x^{-2s}\mathrm{d}s\hfill \\ \hfill & =G_{0,2}^{2,0}\left(x^2\left|\begin{array}{c}-\\ {\displaystyle \frac{\nu }{2}},-{\displaystyle \frac{\nu }{2}}\end{array}\right.\right),\hfill \end{array}$$

(45)

where $\left(c\right)$ is a suitable Bromwich path separating the poles of the integral, which may denote different paths. In another form, it reads [11] [I, p.331]

$$2K_{\nu }\left(2x\right)={\displaystyle \frac{1}{2\pi i}}{\int }_{\left(c\right)}\Gamma \left(s\right)\Gamma \left(s-\nu \right)x^{\nu -2s}\mathrm{d}s.$$

Proposition 2.

The Hecke integral, the left-hand side of Equation (44), is the Mellin transform pair of the K-Bessel function ($\nu =\alpha -\beta$)

$$\begin{array}{cc}\hfill & {\int }_0^{\infty }{t}^{\beta -\alpha }{e}^{-{\displaystyle \frac{a}{t}}-bt}{\displaystyle \frac{\mathrm{d}t}{t}}\hfill \\ \hfill & ={\left({\displaystyle \frac{b}{a}}\right)}^{{\displaystyle \frac{\nu }{2}}}2K_{\nu }\left(2\sqrt{ab}\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{2\pi i}}{\int }_{\left(c\right)}\Gamma \left(s+\alpha \right)\Gamma \left(s+\beta \right)a^{-s-\alpha }{b}^{-s-\beta }\mathrm{d}s.\hfill \end{array}$$

(46)

Hardy’s formula is a special case

$$\begin{array}{cc}\hfill {\int }_0^{\infty }{t}^{\mu }{e}^{-{\displaystyle \frac{z^2}{t}}-t}{\displaystyle \frac{\mathrm{d}t}{t}}& =z^{\mu }2K_{\mu }\left(2z\right)=2{\psi }_{2\mu -1}\left(z\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{2\pi i}}{\int }_{\left(c\right)}\Gamma \left(s+\alpha \right)\Gamma \left(s+\alpha +\mu \right)z^{-2s-2\alpha }\mathrm{d}s.\hfill \end{array}$$

Proof. 

The right-hand side of Equation (44) becomes

$$a^{-\nu }{\left(\sqrt{ab}\right)}^{\nu }{\displaystyle \frac{1}{2\pi i}}{\int }_{\left(c^{\prime }\right)}\Gamma \left(z\right)\Gamma \left(z-\nu \right){\left(\sqrt{ab}\right)}^{-2z+\nu }\mathrm{d}s,$$

and so it is ${\left({\displaystyle \frac{b}{a}}\right)}^{{\displaystyle \frac{\nu }{2}}}2K_{\nu }\left(2\sqrt{ab}\right)$, which is the left-hand side of Equation (44) in view of Equation (23) with $p=1$, $\alpha =b$, $\beta =a$, $s=-\nu =\beta -\alpha$. □

Ref. [62] [pp. 55–58] gave a proof of the identity

$$\begin{array}{cc}\hfill & {\zeta }^{{\displaystyle \frac{\alpha +\beta }{2}}}{\int }_0^{\infty }{t}^{\beta -\alpha }{e}^{-{\displaystyle \frac{a{\zeta }^{1/2}}{t}}-b{\zeta }^{1/2}t}{\displaystyle \frac{\mathrm{d}t}{t}}\hfill \\ \hfill & =\mathsf{\Psi }\left(\zeta \right)\hfill \\ \hfill & ={\displaystyle \frac{1}{2\pi i}}{\int }_{\left(c\right)}\Gamma \left(s+\alpha \right)\Gamma \left(s+\beta \right)a^{-s-\alpha }{b}^{-s-\beta }{\zeta }^{-s}\mathrm{d}s.\hfill \end{array}$$

(47)

Because of the presence of the additional variable, the proof is rather delicate. Hecke’s formula is the special case with $\zeta =1$. Hecke’s integral can be found on [11] [p. 146, (29)] and [12] [p. 85].

It can be seen that the Hecke transform still lies in the world of K-Bessel functions (Fourier–Bessel expansions), and so does [57], and only in Maass [56] did the Fourier–Whittaker expansions rose out.

The G-function hierarchy [2] [p. 151, (3.2.3)] extended from Kuzumaki [31] reads

$$\begin{array}{cc}\hfill & G_{0,1}^{1,0}\left(z\left|\begin{array}{c}-\\ \mu \end{array}\right.\right)=\sqrt{{\displaystyle \frac{2}{\pi }}}{z}^{\mu +{\displaystyle \frac{1}{2}}}{K}_{\pm {\displaystyle \frac{1}{2}}}\left(z\right)=z^{\mu }{e}^{-z}\hfill \\ \hfill & \searrow \hfill \\ \hfill & G_{0,2}^{2,0}\left(z^2\left|\begin{array}{c}-\\ {\displaystyle \frac{\mu }{2}},-{\displaystyle \frac{\mu }{2}}\end{array}\right.\right)=2K_{\mu }\left(2z\right)=\sqrt{\pi }{e}^{-2z}{\left(4z\right)}^{\mu }\mathsf{\Psi }\left(\mu +{\displaystyle \frac{1}{2}},2\mu +1;4z\right)\hfill \\ \hfill & =\sqrt{{\displaystyle \frac{\pi }{z}}}{W}_{0,\mu }\left(4z\right)\hfill \\ \hfill & =2\sqrt{\pi }{e}^{2z}{G}_{1,2}^{2,0}\left(4z\left|\begin{array}{c}{\displaystyle \frac{1}{2}}\\ \mu ,-\mu \end{array}\right.\right)\hfill \\ \hfill & ={\displaystyle \frac{2\sqrt{\pi }{e}^{-2z}}{\Gamma \left(\mu +\frac{1}{2}\right)\Gamma \left(-\mu +\frac{1}{2}\right)}}{G}_{1,2}^{2,1}\left(4z\left|\begin{array}{c}{\displaystyle \frac{1}{2}}\\ \mu ,-\mu \end{array}\right.\right)\hfill \\ \hfill & \searrow \hfill \\ \hfill & W_{\varkappa ,\mu }\left(2z\right)=z^{\mu +{\displaystyle \frac{1}{2}}}W\left(z;\mu +{\displaystyle \frac{1}{2}}+\varkappa ,\mu +{\displaystyle \frac{1}{2}}-\varkappa \right)\hfill \\ \hfill & ={\displaystyle \frac{{\left(2z\right)}^{\frac{1}{2}}}{\Gamma \left(\frac{1}{2}+\mu -\varkappa \right)\Gamma \left(\frac{1}{2}-\mu +\varkappa \right)}}{G}_{1,2}^{2,1}\left(2z\left|\begin{array}{c}\begin{array}{c}{\displaystyle \frac{1}{2}}+\varkappa \\ \mu ,-\mu \end{array}\end{array}\right.\right)e^{-z}\hfill \\ \hfill & \searrow \hfill \\ \hfill & \mathrm{The}\mathrm{Main}\mathrm{formula}H,\hfill \end{array}$$

(48)

where the arrow ↘ indicates the uplift. The second equality may be replaced by Equation (45).

To indicate the counterpart of the Hardy transform in finite fields. We recall the Kloosterman sum defined by [67] as

$$S(m,n;c)=\sum _{\begin{array}{c}d\left(\mathrm{mod}c\right)\\ (d,c)=1\end{array}}{e}^{2\pi i{\displaystyle \frac{m{d}^{*}+nd}{c}}},$$

where $d^{*}$ is the inverse of d mod c. Bellman [24] [(10.1)] states a finite field analog of the Voronoĭ Equation (25), which is an analog of the Kloosterman sum

$$S(q,x^{2}q;c)=\sum _{d=1}^{p-1}{e}^{-2\pi i{\displaystyle \frac{q{d}^{*}+x^{2}qd}{p}}},$$

where p is a prime.

2. Chowla–Selberg Integral Formula for a Non-Holomorphic Eisenstein Series w.r.t. the Hilbert Modular Group

In this section we dwell on Asai [4] (in conjunction with [2]), which contains two remarkable achievements. One is the discovery of a function analogous to the Dedekind eta-function in the Gaussian field F for which he used the second equality of Equation (46) (inverse Heaviside integral) and Weil’s method of using the product of the Dedekind zeta functions ${\zeta }_F\left(s\right){\zeta }_F(s+1)$.

The other is the Kronecker limit formula for the non-holomorphic Eisenstein series $E(s,\mathit{z})$ of the Hilbert modular group ${\mathrm{SL}}_2\left(\mathfrak{o}\right)$. Here $\mathfrak{o}={\mathfrak{o}}_F$ is the ring of integers in F, where F is an arbitrary algebraic number field (for simplicity, the class number $=1$ is assumed). Asai describes his approach as follows: Hecke [63], who originally studied these problems, did seek after “a function analogous to $log\eta \left(z\right)$”, though we choose the simpler way to seek after a function analogous to $log\left|\eta \right(z\left)\right|$ by contrast.

Asai established a way of deducing the Kronecker limit formula from the Fourier–Bessel expansion [4] [(13)], which he derived from the Hecke gamma transform Equation (1), the theta transformation formula and the Hecke transform (the first equality in Equation (46)). Here the theta function is the general theta function of Hecke. Then he deduced the functional equation for $E(s,\mathit{z})$ from that of the Dedekind zeta function and the symmetry of the K-Bessel function.

Let F be an algebraic number field of degree $n=r_1+2r_2$, where $r_1$ resp. $2r_2$ indicates the real resp. imaginary conjugates. For every $\theta \in F$, $\theta \to {\theta }^{\left(j\right)}$ denotes the conjugation map which sends the imaginary ones to ${\theta }^{\left(j\right)}={\theta }^{(j+r_1)}$ for $r_1+1\le j\le r_1+r_2$. The functional equation for the Dedekind zeta function ${\zeta }_F\left(s\right)$ reads

$$A^{-s}{\Gamma }^{r_1}\left({\displaystyle \frac{s}{2}}\right){\Gamma }^{r_2}\left(s\right){\zeta }_F\left(s\right)=A^{-(1-s)}{\Gamma }^{r_1}\left({\displaystyle \frac{1-s}{2}}\right){\Gamma }^{r_2}(1-s){\zeta }_F(1-s),$$

(49)

where

$$A={\displaystyle \frac{2^{r_2}{\pi }^{\frac{\chi }{2}}}{\sqrt{\Delta }}}.$$

Definition 1.

Let ${\mathcal{H}}_q$ denote the quarternion upper half-plane $(z=\left(\begin{array}{cc}x& -y\\ y& x\end{array}\right),y>0)$.

Let ${\mathcal{H}}_n={\mathcal{H}}^{r_1}\times {{\mathcal{H}}_q}^{r_2}$ be the upper half-space corresponding to F.

${\mathcal{H}}_n\ni \mathit{z}=\left(z_j\right)$, where

$$\begin{array}{cc}\hfill & \left\{\begin{array}{cc}{z}_j=x_j+i{y}_j,\hfill & e_j=1,1\le j\le r_1\hfill \\ z_j=\left(\begin{array}{cc}{x}_j& -y_j\\ y_j& x_j\end{array}\right),\hfill & e_j=2,r_1+1\le j\le r_1+r_2,\hfill \end{array}\right.\hfill \\ \hfill & \mathit{y}\left(\mathit{z}\right)=\left(y\left(z_j\right)\right),y_j=y\left(z_j\right)>0,N\mathit{y}\left(\mathit{z}\right)=\prod _{j=1}^{r_1+r_2}y{\left(z_j\right)}^{e_j},\hfill \\ \hfill & \mathit{x}\left(\mathit{z}\right)=\left(x\left(z_j\right)\right),x_j=x\left(z_j\right).\hfill \end{array}$$

(50)

We formally write $\mathit{z}=\mathit{x}+\mathit{i}\mathit{y}$ to indicate Equation (50). In the case where F is purely imaginary, the number of complex conjugates is viewed as $r_1$ above.

The group acting on ${\mathcal{H}}_n$ is $G={\mathrm{SL}}_2{\left(\mathbb{R}\right)}^{r_1}\times {\mathrm{SL}}_2{\left(\mathbb{C}\right)}^{r_2}$. For $G\ni \sigma =\left({\sigma }_j\right)$, ${\sigma }_j=\left(\begin{array}{cc}{\alpha }_j& {\beta }_j\\ {\gamma }_j& {\delta }_j\end{array}\right)$, the action is defined by $\sigma \mathit{z}=\left({\sigma }_j{z}_j\right)$, where

$${\sigma }_j{z}_j={\displaystyle \frac{{\alpha }_j{z}_j+{\beta }_j}{{\gamma }_j{z}_j+{\delta }_j}},$$

and where for $r_1+1\le j\le r_1+r_2$, ${\alpha }_j$ etc. are embedded as $\left(\begin{array}{cc}{\alpha }_j& \\ & {\alpha }_j\end{array}\right)$ and ${\alpha }_j{z}_j$ are matrix products.

$\Gamma ={\mathrm{SL}}_2\left(\mathfrak{o}\right)\subset G$ is its discontinuous subgroup under the identification ${\sigma }_j={\sigma }^{\left(j\right)}=\left(\begin{array}{cc}{\alpha }^{\left(j\right)}& {\beta }^{\left(j\right)}\\ {\gamma }^{\left(j\right)}& {\delta }^{\left(j\right)}\end{array}\right)$ for $\left(\begin{array}{cc}\alpha & \beta \\ \gamma & \delta \end{array}\right)\in \Gamma$; $y\left({\sigma }^{\left(j\right)}{z}_j\right)=y({\gamma }^{\left(j\right)},{\delta }^{\left(j\right)};z_j)$ holds, so that

$$\mathit{y}\left(\sigma \mathit{z}\right)=\mathit{y}(\mathsf{\gamma },\mathsf{\delta };\mathit{z}).$$

${\Gamma }_1$ is the subgroup of Γ given by

$${\Gamma }_1=\{\sigma =\left(\begin{array}{cc}\alpha & \beta \\ 0& \delta \end{array}\right)\in \Gamma \};$$

(51)

Δ is the absolute value of the discriminant of F: $\Delta =N{\mathfrak{o}}^{*}={\left|N\omega \right|}^{-1}$, where ${\mathfrak{o}}^{*}=\left(\omega \right)$ is the inverse different.

We introduce generalizations of the real analytic Eisenstein Equation (28) and the EES Equation (18) in the following definition. To introduce a generalization of (EES), we deviate from Asai and give some notation. For every $\mu ,\nu \in F$ with $(\mu ,\nu )\ne (0,0)$ and every $\mathit{z}=\left(z_j\right)\in {\mathcal{H}}_n$, we define

$$\tilde{y}({\mu }^{\left(j\right)},{\nu }^{\left(j\right)};z_j)={\displaystyle \frac{1}{|{\mu }^{\left(j\right)}+{\nu }^{\left(j\right)}{x}_j{|}^2+{\left|{\nu }^{\left(j\right)}{y}_j\right|}^2}}.$$

(52)

Asai introduced

$$y({\mu }^{\left(j\right)},{\nu }^{\left(j\right)};z_j)=y_j\tilde{y}({\mu }^{\left(j\right)},{\nu }^{\left(j\right)};z_j)={\displaystyle \frac{y_j}{|{\mu }^{\left(j\right)}+{\nu }^{\left(j\right)}{x}_j{|}^2+{\left|{\nu }^{\left(j\right)}{y}_j\right|}^2}}$$

and

$$\mathit{y}(\mu ,\nu ;\mathit{z})=\left(y({\mu }^{\left(j\right)},{\nu }^{\left(j\right)};z_j)\right),N\mathit{y}(\mu ,\nu ;\mathit{z})=\prod _{j=1}^{r_1+r_2}y{({\mu }^{\left(j\right)},{\nu }^{\left(j\right)};z_j)}^{e_j},$$

which reads in conjunction with Equation (52)

$$\tilde{\mathit{y}}(\mu ,\nu ;\mathit{z})=\left(\tilde{y}({\mu }^{\left(j\right)},{\nu }^{\left(j\right)};z_j)\right),N\tilde{\mathit{y}}(\mu ,\nu ;\mathit{z})=\prod _{j=1}^{r_1+r_2}\tilde{y}{({\mu }^{\left(j\right)},{\nu }^{\left(j\right)};z_j)}^{e_j}.$$

It follows that

$$N\mathit{y}(\mu ,\nu ;\mathit{z})=N\mathit{y}\left(\mathit{z}\right)N\tilde{\mathit{y}}(\mu ,\nu ;\mathit{z})=\prod _{j=1}^{r_1+r_2}{y}_j^{e_j}\prod _{j=1}^{r_1+r_2}\tilde{y}{({\mu }^{\left(j\right)},{\nu }^{\left(j\right)};z_j)}^{e_j}.$$

Definition 2.

A generalization of the Eisenstein–Maass Equation (28) is

$$E(s,\mathit{z})=\sum _{\sigma \in {\Gamma }_1\setminus \Gamma }N\mathit{y}{\left(\sigma \mathit{z}\right)}^s.$$

and that of (EES) is

$${\tilde{\zeta }}_{{\mathfrak{o}}^2}(s,\mathit{z})={\sum _{\mu ,\nu }}^{\prime }N\tilde{\mathit{y}}{(\mu ,\nu ;\mathit{z})}^s={\sum _{\mu ,\nu }}^{\prime }\prod _{j=1}^{r_1+r_2}{\left({\displaystyle \frac{1}{|{\mu }^{\left(j\right)}+{\nu }^{\left(j\right)}{x}_j{|}^2+{\left|{\nu }^{\left(j\right)}{y}_j\right|}^2}}\right)}^s,$$

where the prime on the summation sign means that $(\mu ,\nu )$ runs over all non-associated pairs $(\mu ,\nu )\in {\mathfrak{o}}^2$ except $(0,0)$, and where two pairs $(\mu ,\nu )$ and $({\mu }_1,{\nu }_1)$ are called associated if both relations $\mu x=\epsilon \mu$ and ${\nu }_1=\epsilon \nu$ hold with the same unit ε of $\mathfrak{o}$. The real-analytic Eisenstein series (30) is generalized to

$$E^{*}(s,\mathit{z})={\sum _{\mu ,\nu }}^{\prime }N\mathit{y}{(\mu ,\nu ;\mathit{z})}^s=N\mathit{y}{\left(\mathit{z}\right)}^s{\tilde{\zeta }}_{{\mathfrak{o}}^2}(s,\mathit{z}).$$

Then as a reminiscent to Equation (29),

$$\begin{array}{cc}\hfill {\zeta }_F\left(2s\right)E(s,\mathit{z})& =E^{*}(s,\mathit{z})=N\mathit{y}{\left(\mathit{z}\right)}^s{\tilde{\zeta }}_{{\mathfrak{o}}^2}(s,\mathit{z})\hfill \\ \hfill & =N\mathit{y}{\left(\mathit{z}\right)}^s{\sum _{\mu ,\nu }}^{\prime }N\tilde{\mathit{y}}(\mu ,\nu ;\mathit{z})=N\mathit{y}{\left(\mathit{z}\right)}^s{\sum _{\mu ,\nu }}^{\prime }\prod _{j=1}^{r_1+r_2}\tilde{y}{({\mu }^{\left(j\right)},{\nu }^{\left(j\right)};z_j)}^{\left(e_j\right)s},\hfill \end{array}$$

Theorem 3

([4] (13)). The Chowla–Selberg integral formula

$$\begin{array}{cc}\hfill E(s,\mathit{z})& =N\mathit{y}{\left(\mathit{z}\right)}^s{\zeta }_F\left(2s\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{\sqrt{\Delta }}}N\mathit{y}{\left(\mathit{z}\right)}^{1-s}{\left({\displaystyle \frac{\sqrt{\pi }\Gamma (s-\frac{1}{2})}{\Gamma \left(s\right)}}\right)}^{r_1}{\left({\displaystyle \frac{\pi \Gamma (2s-1)}{\Gamma \left(2s\right)}}\right)}^{r_2}{\zeta }_F(2s-1)\hfill \\ \hfill & +{\displaystyle \frac{2^{r_1+r_2}}{{\sqrt{\Delta }}^s}}N\mathit{y}{\left(\mathit{z}\right)}^{{\displaystyle \frac{1}{2}}}{\left({\displaystyle \frac{{\pi }^s}{\Gamma \left(s\right)}}\right)}^{r_1}{\left({\displaystyle \frac{{\left(2\pi \right)}^{2s}}{\Gamma \left(2s\right)}}\right)}^{r_2}\sum _{\begin{array}{c}{(\mu ,\nu )}^{\prime }\\ \mu \nu \ne 0\end{array}}{\left|{\displaystyle \frac{N\nu }{N\mu }}\right|}^{s-{\displaystyle \frac{1}{2}}}{e}^{2\pi iR\left(\mu \nu \omega \mathit{x}\right)}\hfill \\ \hfill & \times \prod _{j=1}^{r_1+r_2}{K}_{e_j(s-{\displaystyle \frac{1}{2}})}\left(2e_j\pi \left|{\left(\mu \nu \omega \right)}^{\left(i\right)}\right|y_j\right)\hfill \end{array}$$

holds true, where

$$R\left(\mu \nu \omega \mathit{x}\right)=\sum _{j=1}^{r_1+r_2}{e}_j\mathrm{Re}\left({\left(\mu \nu \omega \right)}^{\left(j\right)}{x}_j\right)$$

(53)

and ${(\mu ,\nu )}^{\prime }$ is a class of pairs $(\mu ,\nu )\in \mathfrak{o}\times \mathfrak{o}$ under the equivalence relation $(\mu ,\nu )\sim (\mu \epsilon ,\nu {\epsilon }^{-1})$ for a unit ε.

Proof. 

Proof goes in the spirit of, and parallel to, the second Proof of Theorem 1 (cf. [35]) with some more complications. By the dissection

$$\sum _{(\mu ,\nu )\ne (0,0)}=\sum _{\begin{array}{c}0\ne \left(\mu \right)\subset \mathfrak{o}\\ \nu =0\end{array}}+\sum _{0\ne \left(\nu \right)\subset \mathfrak{o}}\sum _{\mu \in \mathfrak{o}},$$

we find that

$${\tilde{\zeta }}_{{\mathfrak{o}}^2}(s,\mathit{z})=\sum _{0\ne \left(\mu \right)\subset \mathfrak{o}}\tilde{\mathit{y}}{(\mu ,0,\mathit{z})}^s+S_2={\zeta }_F\left(2s\right)+S_2,$$

(54)

where

$$\begin{array}{c}\hfill S_2=\sum _{0\ne \left(\nu \right)\subset \mathfrak{o}}\sum _{\mu \in \mathfrak{o}}\prod _{j=1}^{r_1+r_2}{\displaystyle \frac{{\left(e_j\pi \right)}^{e_{j}s}}{\Gamma \left(e_{j}s\right)}}{I}_j(s,\mathit{z})\end{array}$$

(55)

and where

$$\begin{array}{ccc}\hfill I_j(s,\mathit{z})& =& {\int }_0^{\infty }{t}^{e_{j}s}{e}^{-e_j\pi t_j\tilde{y}{({\mu }^{\left(i\right)},{\nu }^{\left(i\right)},z_j)}^{-1}}{\displaystyle \frac{\mathrm{d}{t}_j}{t_j}},\hfill \\ \hfill \sum _{\mu \in \mathfrak{o}}\prod _j{I}_j(s,\mathit{z})& =& {\int }_0^{\infty }\dots {\int }_0^{\infty }\prod _j{t}_j^{e_{j}s}{e}^{-\pi \left({\sum }_j{e}_j{t}_j{y}_j^2\right|{\nu }^{\left(j\right)}{|}^2)}\theta \left(\mathit{t}\right){\displaystyle \frac{\mathrm{d}{t}_j}{t_j}}.\hfill \end{array}$$

(56)

Here $\theta \left(\mathit{t}\right)$ is Hecke’s theta-funtion

$$\theta \left(\mathit{t}\right)=\sum _{\mu \in \mathfrak{o}}{e}^{-\pi \left({\sum }_j{e}_j{t}_j\right|{\mu }^{\left(j\right)}+{\nu }^{\left(j\right)}{x}_j{|}^2)},$$

which satisfies the transformation formula

$$\begin{array}{cc}\theta \left(\mathit{t}\right)\hfill & ={\sqrt{\Delta }}^{-1}{\prod }_j{t}_j^{-{\displaystyle \frac{e_j}{2}}}\sum _{{\mu }_1\in {\mathfrak{o}}^{*}}{e}^{-\pi {\sum }_j{e}_j{t}_j^{-1}{\left|{\mu }_1^{\left(j\right)}\right|}^2+2\pi iR\left({\mu }_1\nu \mathit{x}\right)}\hfill \\ \hfill & ={\sqrt{\Delta }}^{-1}{\prod }_j{t}_j^{-{\displaystyle \frac{e_j}{2}}}+{\sqrt{\Delta }}^{-1}{\prod }_j{t}_j^{-{\displaystyle \frac{e_j}{2}}}\sum _{0\ne {\mu }_1\in {\mathfrak{o}}^{*}}{e}^{-\pi {\sum }_j{e}_j{t}_j^{-1}{\left|{\mu }_1^{\left(j\right)}\right|}^2+2\pi iR\left({\mu }_1\nu \mathit{x}\right)}\hfill \end{array}$$

(57)

where R is defined in Equation (53).

Substituting Equation (57) in the second formula in Equation (56), we find that the first term Equation (57) contributes

$$\begin{array}{cc}\hfill & {\sqrt{\Delta }}^{-1}{\int }_0^{\infty }\dots {\int }_0^{\infty }\prod _j{t}_j^{e_j\left(s-{\displaystyle \frac{1}{2}}\right)}{e}^{-\pi ({\sum }_j{e}_j{y}_j^2|{\nu }^{\left(j\right)}{|}^2)t_j}{\displaystyle \frac{\mathrm{d}{t}_j}{t_j}}\hfill \\ \hfill & ={\sqrt{\Delta }}^{-1}\prod _j\Gamma \left(e_j\left(s-{\displaystyle \frac{1}{2}}\right)\right)\left({\left(\pi e_j{y}_j^2{\left|{\nu }^{\left(j\right)}\right|}^2\right)}^{-e_j\left(s-{\displaystyle \frac{1}{2}}\right)}\right)\hfill \\ \hfill & ={\sqrt{\Delta }}^{-1}\prod _j\Gamma \left(e_j\left(s-{\displaystyle \frac{1}{2}}\right)\right){\pi }^{n\left({\displaystyle \frac{1}{2}}-s\right)}{2}^{r_2\left(1-2s\right)}N\mathit{y}{\left(\mathit{z}\right)}^{1-2s}{\left|\nu \right|}^{1-2s}\hfill \end{array}$$

(58)

since

$$\prod _j{\left(y_j^2{\left|{\nu }^{\left(j\right)}\right|}^2\right)}^{-e_j\left(s-{\displaystyle \frac{1}{2}}\right)}=N\mathit{y}{\left(\mathit{z}\right)}^{1-2s}{\left|\nu \right|}^{1-2s}.$$

Hence that part of $S_2$ in Equation (55) corresponding to Equation (58) is

$$\begin{array}{cc}\hfill & S_2^{\left(1\right)}:=\sum _{0\ne \left(\nu \right)\subset \mathfrak{o}}\sum _{{\mu }_1=0}\prod _{j=1}^{r_1+r_2}{\displaystyle \frac{{\left(e_j\pi \right)}^{e_{j}s}}{\Gamma \left(e_{j}s\right)}}{I}_j(s,\mathit{z})\hfill \\ \hfill & ={\sqrt{\Delta }}^{-1}\prod _j{\displaystyle \frac{\Gamma \left(e_j\left(s-\frac{1}{2}\right)\right)}{\Gamma \left(e_{j}s\right)}}{2}^{r_{2}s}{\pi }^{ns}{2}^{r_2(1-2s)}{\pi }^{n(1/2-s)}N\mathit{y}{\left(\mathit{z}\right)}^{1-2s}\sum _{0\ne \left(\nu \right)\subset \mathfrak{o}}{\left|\nu \right|}^{1-2s}\hfill \\ \hfill & ={\sqrt{\Delta }}^{-1}N\mathit{y}{\left(\mathit{z}\right)}^{1-2s}{\left({\displaystyle \frac{\sqrt{\pi }\Gamma \left(\left(s-\frac{1}{2}\right)\right)}{\Gamma \left(s\right)}}\right)}^{r_1}{\left({\displaystyle \frac{2\pi \Gamma \left(2\pi \left(2s-1\right)\right)}{\Gamma \left(2s\right)}}\right)}^{r_2}{\zeta }_F(2s-1).\hfill \end{array}$$

(59)

Now what remains is that part of $S_2$ which contains the second term in Equation (57)

$$\begin{array}{cc}\hfill & S_2^{\left(2\right)}:=\sum _{0\ne \left(\nu \right)\subset \mathfrak{o}}\sum _{{\mu }_1\ne 0}\prod _{j=1}^{r_1+r_2}{\displaystyle \frac{{\left(e_j\pi \right)}^{e_{j}s}}{\Gamma \left(e_{j}s\right)}}{I}_j(s,\mathit{z})\hfill \\ \hfill & ={\sqrt{\Delta }}^{-1}{\prod }_j{\displaystyle \frac{{\left(e_j\pi \right)}^{e_{j}s}}{\Gamma \left(e_{j}s\right)}}\sum _{{(\mu ,\nu )}^{\prime }}{e}^{2\pi iR\left(\mu \nu \omega \mathit{x}\right)}{\prod }_j{H}_{\mu ,\nu }^{\left(j\right)}\hfill \end{array}$$

on expressing the sum ${\sum }_{0\ne \left(\nu \right)\subset \mathfrak{o}}{\sum }_{{\mu }_1\ne 0}$ as ${\sum }_{\begin{array}{c}{(\mu ,\nu )}^{\prime },\mu \nu \ne 0\\ {\mu }_1=\mu \omega \end{array}}$ or ${\sum }_{{(\mu ,\nu )}^{\prime }}$ as stated in the theorem. Here

$$H_{\mu ,\nu }^{\left(j\right)}={\int }_0^{\infty }{t}_j^{e_j\left(s-{\displaystyle \frac{1}{2}}\right)}{e}^{-e_j\pi \left(y_j^2|{\nu }^{\left(j\right)}{|}^2{t}_j+{\left|{\mu }^{\left(j\right)}{\omega }^{\left(j\right)}\right|}^2{t}_j^{-1}\right)}{\displaystyle \frac{\mathrm{d}{t}_j}{t_j}},$$

which is the Hecke transform in Proposition 2. Hence

$$H_{\mu ,\nu }^{\left(j\right)}={\left|{\displaystyle \frac{{\mu }^{\left(j\right)}{\omega }^{\left(j\right)}}{{\nu }^{\left(j\right)}{y}_j}}\right|}^{e_j\left(s-{\displaystyle \frac{1}{2}}\right)}2K_{e_j\left(s-{\displaystyle \frac{1}{2}}\right)}\left(2\right|{\mu }^{\left(j\right)}{\nu }^{\left(j\right)}{\omega }^{\left(j\right)}\left|y_j\right),$$

so that

$$\begin{array}{cc}{\prod }_j{H}_{\mu ,\nu }^{\left(j\right)}\hfill & =2^{r_1+r_2}{\left|{\displaystyle \frac{\mu \omega }{\nu N\mathit{y}}}\right|}^{s-{\displaystyle \frac{1}{2}}}{\prod }_j{K}_{e_j\left(s-{\displaystyle \frac{1}{2}}\right)}\left(2\right|{\mu }^{\left(j\right)}{\nu }^{\left(j\right)}{\omega }^{\left(j\right)}\left|y_j\right)\hfill \\ \hfill & =2^{r_1+r_2}{\Delta }^{{\displaystyle \frac{1}{2}}-s}N{\mathit{y}}^{{\displaystyle \frac{1}{2}}-s}{\left|{\displaystyle \frac{\mu }{\nu }}\right|}^{s-{\displaystyle \frac{1}{2}}}{\prod }_j{K}_{e_j\left(s-{\displaystyle \frac{1}{2}}\right)}\left(2\right|{\mu }^{\left(j\right)}{\nu }^{\left(j\right)}{\omega }^{\left(j\right)}\left|y_j\right).\hfill \end{array}$$

(60)

Hence

$$\begin{array}{cc}{S}_2^{\left(2\right)}\hfill & =2^{r_1+r_2}{\Delta }^{-s}N{\mathit{y}}^{{\displaystyle \frac{1}{2}}-s}{\prod }_j{\displaystyle \frac{{\left(e_j\pi \right)}^{e_{j}s}}{\Gamma \left(e_{j}s\right)}}\hfill \\ \hfill & \times \sum _{{(\mu ,\nu )}^{\prime }}{\left|{\displaystyle \frac{\mu }{\nu }}\right|}^{s-{\displaystyle \frac{1}{2}}}{e}^{2\pi iR\left(\mu \nu \omega \mathit{x}\right)}{\prod }_j{K}_{e_j\left(s-{\displaystyle \frac{1}{2}}\right)}\left(2\right|{\mu }^{\left(j\right)}{\nu }^{\left(j\right)}{\omega }^{\left(j\right)}\left|y_j\right).\hfill \end{array}$$

(61)

Thus, substituting Equations (59) and (61) in Equation (54), we obtain the Chowla–Selberg integral formula for ${\tilde{\zeta }}_{{\mathfrak{o}}^2}(s,\mathit{z})$, whence we complete the proof. □

The special case of this is the Chowla–Selberg integral formula for the Eisenstein–Maass Equation (31), cf. [2] [(3.6.8)]

Corollary 1.

Suppose $r_2=0$, so that there is no quaternion upper half-plane and $\mathcal{H}$ is the product of $r_1$ copies of the upper half-plane. In the imaginary quadratic case, we regard $r_1=1$. Then the Chowla–Selberg integral formula (cf. [2] [(3.6.8), (3.6.27)]) reads

$$\begin{array}{cc}\mathcal{E}(s,z)={\displaystyle \frac{1}{2}}\zeta \left(2s\right)E(s,z)\hfill & =y^s\zeta \left(2s\right)+y^{1-s}{\displaystyle \frac{\sqrt{\pi }\Gamma (s-\frac{1}{2})}{\Gamma \left(s\right)}}\zeta (2s-1)\hfill \\ \hfill & +{\displaystyle \frac{4y^{\frac{1}{2}}{\pi }^s}{\Gamma \left(s\right)}}\sum _{n=1}^{\infty }{n}^{s-{\displaystyle \frac{1}{2}}}{\sigma }_{1-2s}\left(n\right)cos2\pi nx{K}_{s-{\displaystyle \frac{1}{2}}}\left(2\pi yn\right).\hfill \end{array}$$

(62)

Proof. 

We specify the data. $\mathfrak{o}=\mathbb{Z}=\langle 1\rangle$ and so $\omega =1$, $\Delta =1$ and ${\tilde{\zeta }}_{{\mathfrak{o}}^2}(s,\mathit{z})={\displaystyle \frac{1}{2}}{\zeta }_{{\mathfrak{o}}^2}(s,\mathit{z})$. □

Corollary 2.

$E(s,\mathit{z})$ satisfies the functional equation similar to Equation (49),

$$A^{-2s}{\Gamma }^{r_1}\left(s\right){\Gamma }^{r_2}\left(2s\right)E(s,\mathit{z})=A^{-(2-2s)}{\Gamma }^{r_1}\left(1-s\right){\Gamma }^{r_2}(2-2s)E(1-s,\mathit{z}).$$

(63)

Proof follows in the same way as that of Equation (20) on using Equation (49). The

Table 2. Correspondence between $E(s,\mathit{z})$ and $E(s,z)$.

Data	$\mathit{E}(\mathit{s},\mathit{z})$	$\mathit{E}(\mathit{s},\mathit{z})$
PID	$\mathfrak{o}=\left(\omega \right)$	$\mathbb{Z}=\left(1\right)$
Mod. gr.	$\Gamma ={\mathrm{SL}}_2\left(\mathfrak{o}\right)$	${\mathrm{SL}}_2\left(\mathbb{Z}\right)$
${\Gamma }_1$	(51)	$\{\pm 1\}$
Factor gr.	${\Gamma }_1\setminus \Gamma$	${\mathrm{PSL}}_2\left(\mathbb{Z}\right)$
UHS	${\mathcal{H}}_n$	$\mathcal{H}$
|discr.|	$\Delta$	1

is the correspondence between $E(s,\mathit{z})$ and $E(s,z)$.

3. Concluding Remarks

As we remarked in the Introduction, this is a starter, and we restrict it to the modular relation with only one gamma factor. The next step is to study the case of the ramified type functional equation with two gamma factors, which is treated in [2] in connection with Maass forms. Even the case with two gamma factors is not touched here, which includes the Dedekind zeta function of a real quadratic field as well as the divisor problem associated with ${\zeta }^2\left(s\right)$. With such cases, there appear some other integral transforms that are often applied, including Koshlyakov and Voronoĭ transforms. Furthermore, there is a problem of unifying the Fourier–Whittaker expansion and the Ewald expansion, including the theory of Harmonic Maass forms. We hope to return to these elsewhere.