A Tapestry of Ideas with Ramanujan’s Formula Woven In

Abstract

Zeta-functions play a fundamental role in many fields where there is a norm or a means to measure distance. They are usually given in the forms of Dirichlet series (additive), and they sometimes possess the Euler product (multiplicative) when the domain in question has a unique factorization property. In applied disciplines, those zeta-functions which satisfy the functional equation but do not have Euler products often appear as a lattice zeta-function or an Epstein zeta-function. In this paper, 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 Eisenstein series of level $2\varkappa$ to the complex variable level s. Naturally, generalized Eisenstein series and Barnes multiple zeta-functions arise, which have affinity to dissections, as they are (semi-) lattice functions. The method of Lewittes (and Chapman) and Kurokawa leads to some limit formulas without absolute value due to Tsukada and others. On the other hand, Komori, Matsumoto and Tsumura make use of the Barnes multiple zeta-functions, proving their modular relation, and they give rise to generalizations of Ramanujan’s formula as the generating zeta-function of ${\sigma }_s\left(n\right)$, the sum-of-divisors function. Lewittes proves similar results for the 2-dimensional case, which holds for all values of s. This in turn implies the eta-transformation formula as the extreme case, and most of the results of Chapman. We shall unify most of these as a tapestry of ideas arising from the merging of additive entity (Dirichlet series) and multiplicative entity (Euler product), especially in the case of limit formulas.

1. Introduction and the Underlying Flow of Ideas

In this paper, by a zeta-function, we mean a certain Dirichlet series that is absolutely convergent in a right half-plane, continued meromorphically over the whole plane (with a possible simple pole at $s=1$, say), and satisfying a certain functional equation with gamma factors. We write $s=\sigma +it$, $\sigma ,t\in \mathbb{R}$. Given a zeta-function $Z\left(s\right)={\sum }_{n=1}^{\infty }{\lambda }_n^{-s}$ analytic at $s=0$, it is common to regard the value $e^{-Z^{\prime }\left(0\right)}$ as the Kronecker limit formula, i.e., the Laurent constant at the simple pole at $s=1$, say. This depends on a heuristic reason that $-Z^{\prime }\left(s\right){{|}_{s=0}={\sum }_{n=1}^{\infty }(log{\lambda }_n){\lambda }_n^{-s}|}_{s=0}={\sum }_{n=1}^{\infty }log{\lambda }_n=log{\prod }_{n=1}^{\infty }{\lambda }_n$, whence formally

$$e^{-Z^{\prime }\left(0\right)}=\prod _{n=1}^{\infty }{\lambda }_n.$$

Cf. [1], pp. 94–131, Chapter III.

Another reason comes from Stark’s intensive work [2,3,4,5] on ”L-functions at $s=1$” in which he made clear that instead of the original Kronecker limit formula at $s=1$, the one at $s=0$ is simpler (in view of a possible functional equation). In Part IV, he gave the Kronecker limit formula.

In [6], (§2.4), we developed the Chowla–Selberg integral formula (or the Fourier–Bessel expansion, which is equivalent to the Hecke functional equation) and the Kronecker limit formula as its consequence for ${\zeta }_{{\mathbb{Z}}^2}(s,\tau )$. Here, ${\zeta }_{{\mathbb{Z}}^2}(s,\tau )$ is the Epstein-type Eisenstein series defined by [6], (1.1.7):

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

(1)

where $\sigma >1,\tau =x+iy\in \mathcal{H}$ and the prime on the summation sign means that $m=n=0$ is excluded. Here, $\mathcal{H}$ is the upper half-plane $\mathcal{H}=\{\tau |\mathrm{Im}\tau >0\}$, and the far-right side of Equation (1) is the Epstein zeta-function of a positive definite quadratic form. In what follows, we always write $\tau =x+iy$ ($y>0$).

Ref. [1], p. 96, Example 7.3, gives

$$\underset{m,n=0}{\overset{\infty }{{\prod }^{\prime }}}{\displaystyle \frac{{|m+n\tau |}^2}{y}}=4{\pi }^{2}y{|\eta \left(\tau \right)|}^4,$$

(2)

where $\eta \left(\tau \right)$ is the Dedekind eta-function, defined as follows:

$$\eta \left(\tau \right)=e^{{\displaystyle \frac{\pi i\tau }{12}}}\prod _{n=1}^{\infty }(1-e^{2\pi in\tau }).$$

Here Equation (2) is the case of the discriminant $-d=1$, the prime on the product sign has the same meaning as with the sum Equation (1), and the zeta-function is not [6], (1.1.7), but the non-holomorphic Eisenstein series

$$y^s{\zeta }_{{\mathbb{Z}}^2}(s,\tau )=\underset{m,n}{\sum ^{\prime }}{\displaystyle \frac{y^s}{{|m+n\tau |}^{2s}}}.$$

This is a typical example of a non-holomorphic automorphic function for which we may employ the results known of the associated Epstein zeta-function.

In this paper, we shall make use of the generating zeta-function for Equation (4) stated in Lemma 1. For general basics on zeta-functions, we refer to [6,7]. As will be seen subsequently, the functional equation Equation (5) is rather involved, save for the case $\alpha$ being an odd integer, and so directly proving modular relations is not simple. We shall examine the papers of [8,9,10,11] from the view point of lattice zeta-functions as an alternative. In Lewittes [8], the Bochner modular relation is deduced by dissection (applied to the whole sum for $G(s,\tau )$) and the Lipschitz summation formula (which itself is a modular relation, applied to the component of $G(s,\tau )$), in Komori et al. [9], the ramified Hecke function equation is proved, and in Kurokawa [10], the Kronecker limit formula without absolute value is deduced as the value at $s=0$, as per Stark (with implicit influence of the functional equation).

Lemma 1.

Consider the product $\phi \left(s\right)$ of two Riemann zeta-functions,

$$\phi \left(s\right)=\phi (s,\alpha )=\zeta \left(s\right)\zeta (s+\alpha )=\sum _{n=1}^{\infty }{\displaystyle \frac{{\sigma }_{-\alpha }\left(n\right)}{n^s}},$$

(3)

where the series is absolutely convergent for $\sigma >{\sigma }_{\phi }:=max\{1,1-\mathrm{Re}\alpha \}$ and

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

(4)

is the sum-of-divisors function. This satisfies the asymmetric functional equation

$$\phi (1-s)=4{\left(2\pi \right)}^{-2s+\alpha }cos{\displaystyle \frac{\pi }{2}}scos{\displaystyle \frac{\pi }{2}}(s-\alpha )\mathsf{\Gamma }\left(s\right)\mathsf{\Gamma }(s-\alpha )\phi (s-\alpha ).$$

(5)

In the case of α being an odd integer, it reduces to the Hecke functional equation

$$\begin{array}{c}\hfill \mathsf{\Gamma }(1-s)\phi (1-s)={\left(2\pi \right)}^{-2s+\alpha +1}{(-1)}^{{\displaystyle \frac{\alpha -1}{2}}}\mathsf{\Gamma }(\alpha +1-s)\phi (s-\alpha ).\end{array}$$

(6)

Equation (6) and the Bochner modular relation in symmetric form

$$\sum _{n=1}^{\infty }{\sigma }_{-\alpha }\left(n\right)e^{-2\pi nx}={(-1)}^{{\displaystyle \frac{\alpha -1}{2}}}{x}^{\alpha -1}\sum _{n=1}^{\infty }{\sigma }_{-\alpha }\left(n\right)e^{-{\displaystyle \frac{2\pi n}{x}}}+{\mathrm{P}}_{\alpha }\left(2\pi x\right),$$

(7)

are equivalent, where for $\alpha =2\varkappa +1>0$, the residual function ${\mathrm{P}}_{2\varkappa +1}\left(2\pi x\right)$ is given by

$$\begin{array}{cc}\hfill {\mathrm{P}}_{2\varkappa +1}\left(2\pi x\right)=& {\displaystyle \frac{{\left(2\pi \right)}^{2\varkappa +1}}{2x}}\sum _{j=0}^{\varkappa +1}{(-1)}^{j+1}{\displaystyle \frac{B_{2j}}{\left(2j\right)!}}{\displaystyle \frac{B_{2\varkappa +2-2j}}{(2\varkappa +2-2j)!}}{x}^{2\varkappa +2-2j}\hfill \\ \hfill & +\left\{\begin{array}{cc}-{\displaystyle \frac{1}{2}}\zeta (2\varkappa +1)\left\{1+{(-1)}^{\varkappa +1}{x}^{2\varkappa }\right\}\hfill & if\varkappa \ge 1,\hfill \\ {\displaystyle \frac{1}{2}}logx\hfill & if\varkappa =0,\hfill \end{array}\right.\hfill \end{array}$$

(8)

and for $0>\alpha =1-2\varkappa$ odd, Equation (7) amounts to

$$\begin{array}{cc}\hfill & \sum _{n=1}^{\infty }{\sigma }_{-\alpha }\left(n\right)e^{-2\pi nx}+{\displaystyle \frac{1}{2}}\zeta \left(\alpha \right)\hfill \\ \hfill & ={(-1)}^{{\displaystyle \frac{\alpha -1}{2}}}{x}^{\alpha -1}\left(\sum _{n=1}^{\infty }{\sigma }_{-\alpha }\left(n\right)e^{-{\displaystyle \frac{2\pi n}{x}}}+{\displaystyle \frac{1}{2}}\zeta \left(\alpha \right)\right)-{\displaystyle \frac{1}{2}}{\delta }_{{\displaystyle \frac{-1-\alpha }{2}}0}{\left(2\pi x\right)}^{-1},\hfill \end{array}$$

(9)

or

$$\begin{array}{cc}\hfill & \sum _{n=1}^{\infty }{\sigma }_{2\varkappa -1}\left(n\right)e^{2\pi in\tau }-{\displaystyle \frac{B_{2\varkappa }}{4\varkappa }}={\tau }^{-2k-2}\left\{\sum _{n=1}^{\infty }{\sigma }_{2\varkappa -1}\left(n\right)e^{-{\displaystyle \frac{2\pi in}{\tau }}}-{\displaystyle \frac{B_{2\varkappa }}{4\varkappa }}\right\}.\hfill \end{array}$$

In the literature, the zeta-values are often expressed by the Bernoulli number, cf. Equation (12) below. E.g., on [12], pp. 97–98, we find

$$\zeta \left(0\right)=-{\displaystyle \frac{1}{2}},\zeta (1-2k)=-{\displaystyle \frac{B_{2k}}{2k}}(k\ge 1),\zeta \left(2k\right)={(-1)}^{k+1}{\displaystyle \frac{2^{2k-1}}{\left(2k\right)!}}{B}_{2k}{\pi }^{2k}.$$

(10)

We remark that the common source of Equations (5) and (10) is the asymmetric functional equation for the Riemann zeta-function:

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

(11)

The nth Bernoulli polynomial $B_n\left(x\right)$ is defined by the Taylor expansion

$${\displaystyle \frac{z{e}^{xz}}{e^z-1}}=\sum _{n=0}^{\infty }{\displaystyle \frac{B_n\left(x\right)}{n!}}{z}^n,|z|<2\pi ,$$

(12)

and the nth Bernoulli number $B_n$ is the value $B_n\left(0\right)$. We agree to use both expressions in Equation (10) interchangeably hereafter.

The transition between the Lambert series and the rapidly convergent series is carried out using Liouville’s formula

$$\sum _{m=1}^{\infty }{\displaystyle \frac{a_m}{e^{mx}-1}}=\sum _{n=1}^{\infty }{b}_n{e}^{-nx},\mathrm{Re}x>0,$$

(13)

where $b_n={\sum }_{d\mid n}{a}_d$ with d running through all positive divisors of n, and the left-side expression is called the Lambert series. It seems that this was first explicitly stated by Koshlyakov [13,14,15], I, p. 138, (10.15).

Equation (7) reads ($k\in \mathbb{N}$)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2}}\zeta (2k+1)+\sum _{n=1}^{\infty }{\displaystyle \frac{n^{-2k-1}}{e^{2\pi xn}-1}}\hfill \\ \hfill & ={(-1)}^k{x}^{2k}\left({\displaystyle \frac{1}{2}}\zeta (2k+1)+\sum _{n=1}^{\infty }{\displaystyle \frac{n^{-2k-1}}{e^{2\pi x^{-1}n}-1}}\right)+{\tilde{\mathrm{P}}}_{2k+1}\left(x\right),\hfill \end{array}$$

(14)

which is Ramanujan’s formula in the Bochner modular relation form, where

$$\begin{array}{cc}\hfill {\tilde{\mathrm{P}}}_{2k+1}\left(x\right)& ={\displaystyle \frac{{\pi }^{2k+1}}{x}}\sum _{j=0}^{k+1}{\displaystyle \frac{{(-1)}^j}{(2j-1)!}}\zeta (1-2j)\zeta (2k-2j+2){\left({\displaystyle \frac{x}{\pi }}\right)}^{2k+2-2j},\hfill \end{array}$$

cf. Equation (8). Through the change of variable

$$\alpha =\pi x>0,\beta ={\displaystyle \frac{\pi }{x}}>0,$$

(15)

the original Ramanujan’s formula (Equation (24)) and its Bochner modular relation form (Equation (14)) transform into each other, which we describe as part of the following theorem.

Theorem 1.

The Bochner modular relation (Equation (7)), which holds for $\mathrm{Re}x>0$ and every odd integer α, entails at one end of the spectrum $\alpha =-(2k+1)$, Ramanujan’s formula (Equation (14)), and at the other end, $\alpha =1-2k$, the automorphy of the general Eisenstein series$E_{1-\alpha }\left(\tau \right)$ in Equation (17), where $1-\alpha \ge 2$ is an even integer:

$$E_{1-\alpha }\left(\tau \right)={\tau }^{\alpha -1}{E}_{1-\alpha }\left(-{\displaystyle \frac{1}{\tau }}\right).$$

At the threshold point $\alpha =1$, it entails automorphy (Equation (32)) of the modular form $\eta \left(\tau \right)$ of the half-integral weight.

Definition 1.

We denote the Fourier series as follows:

$$A(s,\tau )=\sum _{n=1}^{\infty }{\sigma }_{s-1}\left(n\right)e^{2\pi in\tau },$$

(16)

which is continued analytically over the whole plane. We define the generalized Eisenstein series $E_s\left(\tau \right)$ as follows:

$$E_s\left(\tau \right)={\displaystyle \frac{\zeta (1-s)}{2}}+A(s,\tau )={\displaystyle \frac{\zeta (1-s)}{2}}+\sum _{n=1}^{\infty }{\sigma }_{s-1}\left(n\right)e^{2\pi in\tau }.$$

(17)

Summarizing [16], pp. 82–83 or [17], pp. 117–137, Chapter 8, we have the following:

Proposition 1.

Each $\mathbb{C}$-lattice may be viewed as $L({\omega }_1,{\omega }_2)=\mathbb{Z}{\omega }_1\oplus \mathbb{Z}{\omega }_2$ with ${\displaystyle \frac{{\omega }_1}{{\omega }_2}}\in \mathcal{H}$. Each $\mathbb{C}$-lattice up to homothety may be identified with an element of $\mathsf{\Gamma }\setminus \mathcal{H}$, where $\mathsf{\Gamma }=PSL(2,\mathbb{Z})$. We may identify modular functions $\varphi \left(\tau \right)$ of weight $2k$ with some lattice functions $\mathsf{\Phi }({\omega }_1,{\omega }_2)$ of weight $2k$:

$$\varphi \left(\tau \right)={\tau }^{2k}\mathsf{\Phi }(1,\tau ).$$

It follows that the automorphic property is intrinsic to lattice functions (and its equivalent, with the ratio τ of the bases) as well as modular functions.

Equation (1) is a typical example of the lattice function. Another is the classical Eisenstein series [16], p. 83, (12), given as follows:

Lemma 2.

The  Eisenstein series is defined by

$$G_{2k}\left(\tau \right):=\sum _{\begin{array}{c}m,n\in \mathbb{Z}\\ (m,n)\ne (0,0)\end{array}}{\displaystyle \frac{1}{{(m+n\tau )}^{2k}}},$$

which is a modular form of weight $2k$. The Laurent expansion (or q-expansion) reads as follows:

$$G_{2k}\left(\tau \right)=2\zeta \left(2k\right)+{\displaystyle \frac{2{\left(2\pi i\right)}^{2k}}{(2k-1)!}}\sum _{n=1}^{\infty }{\sigma }_{2k-1}\left(n\right)e^{2\pi in\tau },\tau \in \mathcal{H}.$$

(18)

Also, $E_{2k}\left(\tau \right)$ is the constant multiple of this:

$$E_{2k}\left(\tau \right)={\displaystyle \frac{{(-1)}^k(2k-1)!}{2{\left(2\pi \right)}^{2k}}}{G}_{2k}\left(\tau \right).$$

(19)

Here, Equation (19) follows from Equation (11).

We shall show that plenty of the subsequent derived results are inheritance of the q-expansion (Equation (18)) (rather than the Bochner modular relation). By equating the partial fraction expansion for the cotangent function and the polylogarithm function of order 0, we obtain the following (cf. Equation (53)):

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{\tau }}+\sum _{n=1}^{\infty }\left({\displaystyle \frac{1}{n+\tau }}-{\displaystyle \frac{1}{n-\tau }}\right)=\pi cot\pi \tau \hfill \\ \hfill & =-\pi i\left(2{\mathrm{Li}}_0\left(e^{2\pi i\tau }\right)+1\right)=-2\pi i\sum _{n=1}^{\infty }{e}^{2\pi in\tau }-\pi i.\hfill \end{array}$$

(20)

By differentiating this $\varkappa -1$ times, we deduce that

$$F(\varkappa ,\tau )=\sum _{m=-\infty }^{\infty }{\displaystyle \frac{1}{{(m+\tau )}^{\varkappa }}}={\displaystyle \frac{{(-2\pi i)}^{\varkappa }}{(\varkappa -1)!}}\sum _{n=1}^{\infty }{n}^{\varkappa -1}{e}^{2\pi in\tau },$$

(21)

where $F(s,\tau )$ is defined in Equation (40). The q-expansion (Equation (18)) follows by substituting Equation (21) with $\tau$ replaced by $n\tau$ in the following:

$$G_{2k}\left(\tau \right)=2\zeta \left(2k\right)+2\sum _{n=1}^{\infty }\sum _{m=-\infty }^{\infty }{\displaystyle \frac{1}{{(m+n\tau )}^{2k}}}=2\zeta \left(2k\right)+2\sum _{n=1}^{\infty }F(2k,n\tau ).$$

(22)

Definition 2.

Lewittes [8] introduced the generalized Eisenstein series,

$$G(s,\tau )=\underset{m,n=-\infty }{\overset{\infty }{\sum ^{\prime }}}{\displaystyle \frac{1}{{(m+n\tau )}^s}},$$

(23)

absolutely convergent for $\sigma >2$, where the summation sign indicates that $m,n$ range over all integers except $m=n=0$.

Combining Equations (41) and (43), we deduce the following:

$$G(s,\tau )=(1+e^{\pi is})\left(\zeta \left(s\right)+{\displaystyle \frac{{(-2\pi i)}^s}{\mathsf{\Gamma }\left(s\right)}}\left(E_s\left(\tau \right)-{\displaystyle \frac{\zeta (1-s)}{2}}\right)\right).$$

This indicates that $G(s,\tau )$ accommodates both $G_{2k}\left(\tau \right)$ (as a special case) and $E_s\left(\tau \right)$ (as an accommodated element) and suggests that one can derive many results on those Eisenstein series as consequences.

2. Main Results

In view of the raison-d’être for automorphy in Proposition 1, we understand that lattice functions are the other main ingredients. From this point of view, we are to examine the papers of Chapman [18], Komori et al. [9], Kurokawa [10], and Lewittes [8] which are written in complete independence of each other but are all concerned with lattice functions and use some dissection methods. Ref. [18] is concerned with the pseudo-automorphy of $E_k\left(\tau \right)$ for odd integer k in Theorem 2. Save for the Chapman Dedekind sum part, others can be accommodated in [8], and we dwell on those three papers. Lewittes uses the Barnes double zeta-function implicitly (proving the results on the Barnes double zeta-function anew) and gives a generalization of the Bochner modular relation (Theorem 2), while Komori et al. use the Barnes multiple zeta-function and prove the original version (Equation (24)). It is stated for $\alpha >0,\beta >0$ satisfying the relation

$$\alpha \beta ={\pi }^2,$$

and for any positive integer k, as

$$\begin{array}{cc}\hfill & {\alpha }^{-k}\left\{{\displaystyle \frac{1}{2}}\zeta (2k+1)+\sum _{n=1}^{\infty }{\displaystyle \frac{n^{-2k-1}}{e^{2\alpha n}-1}}\right\}\hfill \\ \hfill & ={(-\beta )}^{-k}\left\{{\displaystyle \frac{1}{2}}\zeta (2k+1)+\sum _{n=1}^{\infty }{\displaystyle \frac{n^{-2k-1}}{e^{2\beta n}-1}}\right\}\hfill \\ \hfill & -2^{2k}\sum _{j=0}^{k+1}{(-1)}^j{\displaystyle \frac{B_{2j}}{\left(2j\right)!}}{\displaystyle \frac{B_{2k+2-2j}}{(2k+2-2j)!}}{\alpha }^{k+1-j}{\beta }^j.\hfill \end{array}$$

(24)

As is stated in Section 1, the key point is to extend this as in Lemma 1.

It seems that Komori et al. do not use a dissection, but in their proof of the functional equation, the Atkinson dissection was used. Since the Barnes multiple zeta-functions are semi-lattice functions, they may be accommodated in this category. The Barnes–Hurwitz zeta-function (or multiple zeta-function) is defined in Equation (25) for a basis $\mathit{\omega }=({\omega }_1,\cdots ,{\omega }_r)$ and the associated semi-lattice:

$$\mathsf{\Lambda }=L_{\ge 0}\left(\mathit{\omega }\right)=\left\{\mathsf{\Omega }\right\}={\mathbb{Z}}_{\ge 0}{\omega }_1\oplus \cdots \oplus {\mathbb{Z}}_{\ge 0}{\omega }_r,$$

suppose ${\omega }_1,\cdots ,{\omega }_r,\alpha$ lie in the same side of a line going through the origin. Then, it is defined as follows:

$${\zeta }_r(s,\alpha ,\mathit{\omega })=\sum _{\mathsf{\Omega }\in \mathsf{\Lambda }}{\displaystyle \frac{1}{{(\alpha +\mathsf{\Omega })}^s}}=\sum _{m_1=0}^{\infty }\cdots \sum _{m_r=0}^{\infty }{\displaystyle \frac{1}{{(\alpha +m_1{\omega }_1+\cdots +m_r{\omega }_r)}^s}},$$

(25)

absolutely convergent for $\sigma >r$. It is continued meromorphically over the whole plane and satisfies the functional equation of Hecke type, cf. Section 4. The common procedure is to reduce the lattice zeta-functions to Barnes zeta-functions and apply the known results. This seems to have been done effectively by Hardy and Littlewood and much later by Shintani. The shift in $\alpha$ is rather delicate. For Lewittes’ case, it is 0, but then in the final regularized product results, the required case ${\zeta }_2^{\prime }\left(0,{\displaystyle \frac{1}{2}},(1,\tau +{\displaystyle \frac{1}{2}})\right)$ is not included. Komori et al. incorporated the parameter $a(y_1,\cdots ,y_r)={\omega }_1(1-y_1)+\cdots +{\omega }_1(1-y_r)$, (Equation (61)) in Equation (25). This is convenient to make the case of $y_j=1$ resp. $y_j={\displaystyle \frac{1}{2}}$ accommodated, which presents the case of summation variables being $m_j\ge 1$ resp. $m_j\in \mathbb{Z}+{\displaystyle \frac{1}{2}}$, which in turn amounts to the case of ordinary zeta-functions and the hyperbolic sine case. But again, the required case ${\zeta }_2^{\prime }$ is not included. And, it is included only in Kurokawa’s zeta-function. This suggests the following:

• Conjecture. The theory of ${\zeta }_r\left(s,z+a\left(y\right),\mathit{\omega }\right)$ will cover all the above cases (its special case $r=2$ will suffice).

We are in a position to state the main results (including those from references [8,9,10,18]), in somewhat revised and organized form.

Theorem 2

(Ramanujan formula as per Lewittes). The general Ramanujan formula holds true for all $s\in \mathbb{C}$:

$$A\left(s,-{\displaystyle \frac{1}{\tau }}\right)={\tau }^{s}A(s,\tau )+{\displaystyle \frac{\mathsf{\Gamma }\left(s\right)}{{\left(2\pi i\right)}^s}}({\tau }^s-e^{\pi is})\zeta \left(s\right)+{\displaystyle \frac{\mathsf{\Gamma }\left(s\right)}{{\left(2\pi i\right)}^s}}{\tau }^s(1-e^{\pi is})K(s,\tau ),$$

(26)

where $A(s,\tau )$ is the Fourier series (Equation (16)) and $K(s,\tau )$ is the Barnes double zeta-function in Equation (45). The special case of $s=\alpha$ being an odd integer of Equation (26) is equivalent to the functional equation given in Equation (6) as the Bochner modular relation, where $\phi \left(s\right)=\zeta \left(s\right)\zeta (s+\alpha )$ is defined by Equation (3).

Both Equation (47) and Equation (26) hold for all s values, and each term is analytically continued over the whole plane. Therefore, it follows that $K(s,\tau )$ also has an analytic continuation over the whole plane.

Theorem 2 implies the following:

Corollary 1.

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{(-2\pi i)}^s}{\mathsf{\Gamma }\left(s\right)}}\left(E_s\left(\tau \right)-E_s\left(-{\displaystyle \frac{1}{\tau }}\right){\tau }^{-s}\right)\hfill \\ \hfill & =(1+{\tau }^{-s})\zeta \left(s\right)+(1-{\tau }^{-s}){\displaystyle \frac{\zeta (1-s)}{2}}+(1-e^{\pi is})K(s,\tau ),\hfill \end{array}$$

(27)

which entails

$$\begin{array}{cc}\hfill & \underset{\tau \to x}{lim}\left(E_s\left(\tau \right)-E_s\left(-{\displaystyle \frac{1}{\tau }}\right){\tau }^{-s}\right)\hfill \\ \hfill & ={\displaystyle \frac{\mathsf{\Gamma }\left(s\right)}{{(-2\pi i)}^s}}\left((1+x^{-s})\zeta \left(s\right)+(1-x^{-s}){\displaystyle \frac{\zeta (1-s)}{2}}+(1-e^{\pi is})K(s,x)\right),\hfill \end{array}$$

(28)

for every $x>0$.

Proof. 

For a proof of Equation (28), we note the following. The series for $K(s,\tau )$ in Equation (45) converges uniformly for $\mathrm{Re}\tau \ge \delta$ for each $\delta >0$. Hence, it follows that for $x>0$${lim}_{\tau \to x}K(s,\tau )=K(s,x)$. □

The following corollary provides a further refinement of the elaboration of [18], Theorem 1, given in [19].

Corollary 2

([18]). For odd integer $k\ge 3$ and for every $x>0$,

$$\underset{\tau \to x}{lim}\left(E_k\left(-{\displaystyle \frac{1}{\tau }}\right)-{\tau }^k{E}_k\left(\tau \right)\right)={\displaystyle \frac{(k-1)!}{{\left(2\pi i\right)}^k}}{H}_k\left(x\right),$$

where

$$H_k\left(x\right)=\zeta \left(k\right)(x^k+1)+2x^{k}K(k,x),$$

(29)

which satisfies the transformation formula

$$H_k(1/x)=x^{-k}{H}_k\left(x\right).$$

Further, the closed formula holds true ($a,b\in \mathbb{N},(a,b)=1$):

$$a^{-k}{H}_k(a/b)={\displaystyle \frac{2\zeta (k-1)}{ab}}-i{\mathcal{C}}_a^{b,1}(0,k)-i{\mathcal{C}}_b^{a,1}(0,k),$$

(30)

where

$${\mathcal{C}}_a^{b,1}(0,k)={\displaystyle \frac{1}{a}}\sum _{\mu =1}^{a-1}cot{\displaystyle \frac{\pi b\mu }{a}}{\ell }_k\left({\displaystyle \frac{\mu }{a}}\right),$$

is Chapman’s Dedekind sum.

Proof. 

We denote by $x^{-s}{H}_s\left(x\right)$ the right-hand side of Equation (28) with $\tau$ replaced by $x>0$:

$$x^{-s}{H}_s\left(x\right)=(1+x^{-s})\zeta \left(s\right)+(1-x^{-s}){\displaystyle \frac{\zeta (1-s)}{2}}+(1-e^{\pi is})K(s,x).$$

Then,

$$x^{-s}{H}_s\left({\displaystyle \frac{1}{x}}\right)=(1+x^s)\zeta \left(s\right)+(1-x^s){\displaystyle \frac{\zeta (1-s)}{2}}+(1-e^{\pi is})x^{s}K(s,x),$$

since $K(s,1/x)=e^{\pi is}{x}^{s}L(s,-x)=x^{s}K(s,x)$ based on Equation (46). Hence,

$$H_s\left({\displaystyle \frac{1}{x}}\right)=x^{-s}H(s,x),$$

which entails Equation (29).

Proof of Equation (30) amounts to establishing

$$\begin{array}{cc}\hfill a^{-k}{H}_k(a/b)& =\left({\displaystyle \frac{1}{a}}+{\displaystyle \frac{1}{b}}\right)\zeta \left(k\right)+{\displaystyle \frac{2}{ab}}\zeta (k-1)-{\displaystyle \frac{a+b}{ab}}\zeta \left(k\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{ai}}\sum _{\mu =1}^{a-1}cot{\displaystyle \frac{\pi b\mu }{a}}{\ell }_k\left({\displaystyle \frac{\mu }{a}}\right)+{\displaystyle \frac{1}{bi}}\sum _{\nu =1}^{b-1}cot{\displaystyle \frac{\pi a\nu }{b}}{\ell }_k\left({\displaystyle \frac{\nu }{b}}\right),\hfill \end{array}$$

where ${\ell }_k\left(s\right)$ is the Lerch zeta-function. More computation is needed. □

Corollary 3.

Lewittes’ generalized Ramanujan formula (Equation (26)) in the case of $s=k$ being an integer amounts to

$$A\left(s,-{\displaystyle \frac{1}{\tau }}\right)={\tau }^{s}A(s,\tau )+{\displaystyle \frac{\mathsf{\Gamma }\left(s\right)}{{\left(2\pi i\right)}^s}}({\tau }^s-e^{\pi is})\zeta \left(s\right)-{\displaystyle \frac{{\tau }^s}{{\left(2\pi i\right)}^s(1+e^{\pi is})}}J(s,\tau ),$$

(31)

where on account of Equation (60), we have

$$J(s.\tau )={\int }_C{\displaystyle \frac{w^{s-1}}{(e^w-1)(e^{\tau w}-1)}}\mathrm{d}w=\mathsf{\Gamma }\left(s\right)(e^{2\pi is}-1){\zeta }_2(s,a\left(0\right),(1,\tau )),$$

where C is a Hankel contour in Lemma 3. The special case of Equation (31) with $k=0$ leads to the transformation formula $\eta \left(\tau \right)$ for the Dedekind eta-function.

$$log\eta \left(\tau \right)=log\eta \left(-{\displaystyle \frac{1}{\tau }}\right)-{\displaystyle \frac{1}{2}}log{\displaystyle \frac{\tau }{i}}.$$

(32)

For Equation (32), cf. e.g., [20,21].

In view of Equation (45), Equation (31) is a restatement of Equation (26). Then, Lewittes evaluates $J(k,\tau )$ directly, but we appeal to Equation (63), which gives the evaluation for $J(2\varkappa ,\tau )$ ($\varkappa \in \mathbb{Z}$), and so Corollary 3 leads to Ramanujan’s formula Equation (79). The negative even integer case is also included as stated in Equation (9).

Now, we turn to Kronecker limit formulas without absolute value. We use the q-notation:

$$q_{\tau }=e^{2\pi i\tau },q_z=e^{2\pi iz}.$$

Theorem 3.

We have the limit formulas.

(i) ([22], Proposition 2.3) For the function $F(s,\tau )$ in Equation (40), we have the following:

$$\prod _{m=1}^{\infty }(m+\tau )=e^{-F^{\prime }(0,\tau )}=2sin\pi \tau e^{\pi i(\tau -{\displaystyle \frac{1}{2}})}=\left\{\begin{array}{cc}1-q_{\tau }\hfill & \mathrm{Im}\tau >0\hfill \\ 1-q_{{\displaystyle \frac{1}{\tau }}}\hfill & \mathrm{Im}\tau <0.\hfill \end{array}\right.$$

(33)

(ii) We have

$$A^{\prime }(0,\tau )=\sum _{n=1}^{\infty }a\left(n\right)e^{2\pi im\tau },$$

(34)

where

$$a\left(n\right)=\sum _{d|n}{\displaystyle \frac{logd}{d}},$$

is the arithmetic function studied by Erdös and Zaremba [23] and

$$G^{\prime }(0,\tau )={\zeta }^{\prime }\left(0\right)+A(0,\tau )=-log\eta \left(\tau \right)+{\displaystyle \frac{\pi i\tau }{12}}-{\displaystyle \frac{1}{2}}log2\pi ,$$

(35)

or

$$\underset{m,n=-\infty }{\overset{\infty }{\prod ^{\prime }}}(m+n\tau )=e^{-G^{\prime }(0,\tau )}={\displaystyle \frac{\eta \left(\tau \right)}{\sqrt{2\pi }}}{e}^{{\displaystyle \frac{\pi i\tau }{12}}}.$$

(36)

As a corollary to Theorem 7, we have the zeta-regularization formula corresponding to Equation (2):

Theorem 4

([24]). Let $\tau =x+iy$, $y>0$. Then, the path integral (with imaginary time) ${\tilde{Z}}_{\tau }$ has the expression as the product of determinants (if they exist):

$${\tilde{Z}}_{\tau }=dety\left(-i{\displaystyle \frac{\partial }{\partial t}}-{\displaystyle \frac{\partial }{\partial \theta }}\right)dety\left(-i{\displaystyle \frac{\partial }{\partial t}}+{\displaystyle \frac{\partial }{\partial \theta }}\right),$$

(37)

We have the following explicit formula:

$${\tilde{Z}}_{\tau }=\prod _{k=0}^{\infty }{\left(1+q_{\tau }^{2k+1}\right)}^2\prod _{k=0}^{\infty }{\left(1+{\overline{q_{\tau }}}^{2k+1}\right)}^2.$$

(38)

Proof. 

The eigenvalues ${\mathsf{\Lambda }}_{L_{\pm }}$ of $L_{-}:=-i{\displaystyle \frac{\partial }{\partial t}}-{\displaystyle \frac{\partial }{\partial \theta }}$ resp. $L_{+}:=-i{\displaystyle \frac{\partial }{\partial t}}+{\displaystyle \frac{\partial }{\partial \theta }}$ are $\{m+n\tau \}$ resp. $\{m+n\overline{\tau }\}$, where $m,n\in \mathbb{Z}+{\displaystyle \frac{1}{2}}$. Then, consider the following zeta-function:

$${\zeta }_{\tau }\left(s\right):=\sum _{m,n\in \mathbb{Z}+{\displaystyle \frac{1}{2}}}{\displaystyle \frac{1}{{(m+n\tau )}^s}}=\phi \left(s,{\displaystyle \frac{1}{2}},\left(1,\tau +{\displaystyle \frac{1}{2}}\right)\right),$$

where $\phi$ is defined by Equation (71), and we interpret the determinant of the operators $L_{\pm }$ as the zeta-regularized determinant:

$$det\left(L_{-}\right)=e^{-{\zeta }_{\tau }^{\prime }\left(0\right)},det\left(L_{+}\right)=e^{-{\zeta }_{\overline{\tau }}^{\prime }\left(0\right)}.$$

Hence, Theorem 7 is applicable to give the following:

$$det\left(L_{-}\right)=\prod _{m,n\in \mathbb{Z}+{\displaystyle \frac{1}{2}}}(m+n\tau )=e^{{\zeta }_{\tau }^{\prime }}\left(0\right)=\prod _{n=0}^{\infty }(1+q_{\tau }^{2n+1}),$$

and similarly for $det\left(L_{+}\right)$. Substituting these into Equation (37), we deduce Equation (38). □

As a consequence of Theorem 6 and results from the Barnes double zeta-function, we have the following:

Theorem 5.

The following functional equation holds:

$$\begin{array}{cc}\hfill & -{\displaystyle \frac{\mathsf{\Gamma }\left(s\right)(e^{2\pi is}-1)}{2\pi i}}{\zeta }_2(s,a\left(0\right),(1,\tau ))+{\left(-{\displaystyle \frac{1}{2\pi i}}\right)}^{1-s}\zeta (1-s)+{\tau }^{-1}{\left({\displaystyle \frac{\tau }{2\pi i}}\right)}^{1-s}\zeta (1-s)\hfill \\ \hfill & =\sum _{m=1}^{\infty }\left({\displaystyle \frac{e^{2\pi im\tau }}{1-e^{2\pi im\tau }}}\right)\left({\left(-{\displaystyle \frac{1}{2\pi im}}\right)}^{1-s}-{\left({\displaystyle \frac{1}{2\pi im}}\right)}^{1-s}\right)\hfill \\ \hfill & +{\tau }^{-1}\sum _{m=1}^{\infty }\left({\displaystyle \frac{e^{-2\pi im{\tau }^{-1}}}{1-e^{-2\pi im{\tau }^{-1}}}}\right)\left({\left({\displaystyle \frac{\tau }{2\pi im}}\right)}^{1-s}-{\left(-{\displaystyle \frac{\tau }{2\pi im}}\right)}^{1-s}\right).\hfill \end{array}$$

This may be viewed as a generalization of Ramanujan’s formula (Equation (79)), which is the case of $s=-2\varkappa$ with $0\le \varkappa \in \mathbb{Z}$.

For the derivative, we have the following:

$$\begin{array}{cc}\hfill & {\zeta }_r^{\prime }(0,a(1/2),(1,\omega ))\hfill \\ \hfill & ={\displaystyle \frac{\gamma +2\pi i}{2ßi}}\left(\sum _{m\in \mathbb{Z}\setminus \left\{0\right\}}{\displaystyle \frac{{(-1)}^m}{m}}\left({\displaystyle \frac{e^{\pi im\tau }}{1-e^{2\pi im\tau }}}\right)+\sum _{m\in \mathbb{Z}\setminus \left\{0\right\}}{\displaystyle \frac{{(-1)}^m}{m}}\left({\displaystyle \frac{e^{-\pi im{\tau }^{-1}}}{1-e^{2\pi im{\tau }^{-1}}}}\right)\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{2ßi}}\sum _{m\in \mathbb{Z}\setminus \left\{0\right\}}{\displaystyle \frac{{(-1)}^m}{m}}\left({\displaystyle \frac{e^{\pi im\tau }}{1-e^{2\pi im\tau }}}\right)log{\displaystyle \frac{1}{2\pi m}}\hfill \\ \hfill & +{\displaystyle \frac{1}{2ßi}}\sum _{m\in \mathbb{Z}\setminus \left\{0\right\}}{\displaystyle \frac{{(-1)}^m}{m}}\left({\displaystyle \frac{e^{-\pi im{\tau }^{-1}}}{1-e^{2\pi im{\tau }^{-1}}}}\right)log{\displaystyle \frac{\tau }{2\pi m}},\hfill \end{array}$$

(39)

which leads to a variant of Ramanujan’s formula and the limit formula.

3. Lewittes’ Modular Relation for the Generalized Eisenstein Series

We provide proofs of some of the results stated in Section 2 by slightly modifying the results of [8,18]. It turns out that most of Chapman’s results are consequences of Lewittes’. We mainly state Lewittes’ results and add remarks in square brackets on the corresponding results of Chapman’s.

Proof of Theorem 2.

Proof of equivalence follows from Lemma 1, and Equation (26) follows from Equation (48) in view of ${(-2\pi i)}^s=e^{\pi is}{\left(2\pi i\right)}^s$. Hence, it suffices to prove Equation (48).

Lewittes [8] restricts the argument not as in Equation (72) but as follows:

$$-\pi \le argw<\pi ,0\ne w\in \mathbb{C},-1=e^{-\pi i},$$

The zeta-function is defined as follows:

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

(40)

which is absolutely and uniformly convergent for $\sigma >1$ ($\mathrm{Im}\tau \ne 0$).

Lewittes extracts the cases of $n=0$ from $(\mathrm{i})_K$$m\ge 0,n\ge 0$ and $(\mathrm{i}\mathrm{i}\mathrm{i})_K$$m<0,n\ge 0$ and divides $(\mathrm{i})_K\cup$$(\mathrm{i}\mathrm{i}\mathrm{i})_K$ into three cases: $(\mathrm{i})_L$$m\ge 0,n=0$, $(\mathrm{i}\mathrm{i})_L$$m<0,n=0$, and $(\mathrm{i}\mathrm{i}\mathrm{i})_L$$m\in \mathbb{Z},n>0$. Similarly, the remaining two cases, $(\mathrm{i}\mathrm{i})_K$$m\ge 0,n<0$ and $(\mathrm{i}\mathrm{v})_K$$m<0,n<0$, are unified as $(\mathrm{i}\mathrm{v})_L$$m\in \mathbb{Z},n<0$.

This proves Lewttes’ formula:

$$G(s,\tau )=(1+e^{\pi is})\left(\zeta \left(s\right)+\sum _{n=1}^{\infty }F(s,n\tau )\right),$$

(41)

corresponding to Equation (22), where $G(s,\tau )$ is defined by Equation (23).

Based on the upper half-plane version of the Lipschitz summation formula,

$$\begin{array}{c}\hfill {\displaystyle \frac{{(-2\pi i)}^s}{\mathsf{\Gamma }\left(s\right)}}\sum _{n=0}^{\infty }{(n+w)}^{s-1}{e}^{2\pi i\tau (n+w)}=\sum _{n=-\infty }^{\infty }{\displaystyle \frac{e^{2\pi inw}}{{(\tau +n)}^s}},\end{array}$$

we have

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

Hence, based on the upper half-plane version of Liouville’s formula, we obtain the following:

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

(42)

[This corresponds to [18], (3) in the following form:

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

(43)

]

Hence, substituting Equation (42) into Equation (41) leads to [8], (3)

$$H(s,\tau ):={\displaystyle \frac{G(s,\tau )}{1+e^{\pi is}}}=\zeta \left(s\right)+{\displaystyle \frac{{(-2\pi i)}^s}{\mathsf{\Gamma }\left(s\right)}}A(s,\tau ),$$

(44)

where $A(s,\tau )$ is defined by Equation (16). The series for $A(s,\tau )$ is absolutely and uniformly convergent for $(s,\tau )$ in any compact subset of $\mathbb{C}\times \mathcal{H}$, hence $A(s,\tau )$, and a fortiori, $G(s,\tau )$, have analytic continuation over the whole s-plane. However, $H(s,\tau )$ is a meromorphic function with a simple at $s=1$ with residue 1.

[In [18], $A(k,\tau )=E_k\left(\tau \right)$ for odd integer $k\ge 3$. However, in the original definition given by Kurokawa [11], there is the correction term ${\displaystyle \frac{\zeta (1-s)}{2}}$, which happens to vanish for $s=k$ for an odd integer $\ge 3$. Hence, we are led to define the generalized Eisenstein series by Equation (17). Cf. Corollary 1.]

If we can find the Laurent constant of Equation (23) in closed form, then it is the Kronecker limit formula. Here, we stick to the zeta-regularization and prove Equation (36).

To proceed further, Lewittes introduces another dissection that is almost the same as the Kurokawa dissectionn except for the excludion of two cases of $m=0$ and $n=0$. Here, e.g., $(1)_L$ means Lewittes’ dissection no. (1), and similarly for $(1)_K$ for Kurokawa below.

$(1)_L$$m>0,n>0$, $(1\prime )_L$$m<0,n<0$, $(2)_L$$m<0,n<0$, $(2\prime )_L$$m>0,n<0$, $(3)_L$$m>0,n=0$, $(3\prime )_L$$m<0,n=0$, $(4)_L$$m=0,n>0$, and $(4\prime )_L$$m=0,n<0$, where the term in (n’) is $e^{\pi s}$ times of (n). We use the zeta-functions

$$L(s,\tau )=\sum _{m=-\infty }^{-1}\sum _{n=1}^{\infty }{\displaystyle \frac{1}{{(m+n\tau )}^s}},$$

as well as

$$K(s,\tau )=\sum _{m,n=1}^{\infty }{\displaystyle \frac{1}{{(m+n\tau )}^s}}={\zeta }_2(s,a\left(0\right),(1,\tau )),$$

(45)

where the far-right member indicates the Barnes double zeta-function Equation (25) with $r=2$, $\alpha =a\left(0\right)$.

Note that $(1)_L+$$(1\prime )_L$ gives $(1+e^{\pi is})K(s,\tau )$, $(2)_L+$$(2\prime )_L$ gives $(1+e^{\pi is})L(s,\tau )$, $(3)_L+$$(3\prime )_L$ gives $(1+e^{\pi is})\zeta \left(s\right)$, and $(4)_L+$$(4\prime )_L$ gives $(1+e^{\pi is}){\tau }^{-s}\zeta \left(s\right)$. Substituting these into $G(s,\tau )$ values in Equation (44), we obtain the following:

$$H(s,\tau )=\left(1+{\tau }^{-s}\right)\zeta \left(s\right)+K(s,\tau )+L(s,\tau ).$$

[[18], (2), $F_k\left(\tau \right)$ is $K(k,\tau )$ in Equation (45).]

It follows that

$$K(s,-1/\tau )={\tau }^{s}L(s,\tau ),L(s,-1/\tau )={\tau }^s{e}^{-\pi is}K(s,\tau ).$$

(46)

Therefore, we could apply the method of Section 4, but we partly reproduce Lewittes’ argement combined with the theory of Barnes zeta-functions. Moreover, since $arg(-1/\tau )=\pi -arg\tau$, we have ${\tau }^{-s}=e^{-\pi is}{\tau }^s$, and we have the following transformation formula:

$$H(s,-1/\tau )={\tau }^{s}H(s,\tau )+{\tau }^s(e^{-\pi is}-1)(\zeta \left(s\right)+K(s,\tau )).$$

(47)

Substituting Equation (44) in the above and solving in $A(s,\tau )$, we conclude the following:

$$A\left(s,-{\displaystyle \frac{1}{\tau }}\right)={\tau }^{s}A(s,\tau )+{\displaystyle \frac{\mathsf{\Gamma }\left(s\right)}{{(-2\pi i)}^s}}({\tau }^s{e}^{-\pi is}-1)\zeta \left(s\right)+{\displaystyle \frac{\mathsf{\Gamma }\left(s\right)}{{(-2\pi i)}^s}}{\tau }^s(e^{-\pi is}-1)K(s,\tau ),$$

(48)

which is to be proved. □

Remark 1.

In [25], Kim introduced “two-sided” L-series, called the H-series ($H(s;f)$), which corresponds to the Laurent expansion. The following relation was proved:

$$4\pi sin\left({\displaystyle \frac{\pi s}{2}}\right)A(1-s,\tau )=H(s;f),$$

(49)

where

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

From Equation (49), which corresponds to Equation (44), he deduced the integral representation of $A(s,\tau )$ as the Mellin transform of the Weierstrass zeta-function. This will be touched on elsewhere.

Proof of Corollary 1.

Another proof of Corollary 1 can be given using Equation (42) and Equation (17) and transforming the difference

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

(50)

Equation (50) amounts to the following:

$$\sum _{n=1}^{\infty }\sum _{m=0}^{\infty }{\displaystyle \frac{1}{{(m+n\tau )}^s}}-\sum _{n=-\infty }^0\sum _{m=-\infty }^{-1}{\displaystyle \frac{1}{{(m+n\tau )}^s}}.$$

Extracting the special case with $m=0$ resp. $n=0$, which gives ${\tau }^{-s}\zeta \left(s\right)$ resp. $e^{\pi is}\zeta \left(s\right)$, we have the following:

$$\sum _{n=1}^{\infty }\sum _{m=1}^{\infty }{\displaystyle \frac{1}{{(m+n\tau )}^s}}-e^{\pi is}\sum _{n=1}^{\infty }\sum _{m=1}^{\infty }{\displaystyle \frac{1}{{(m+n\tau )}^s}}=(1-e^{\pi is})K(s,\tau ).$$

This proves Equation (27).

[In the case of Chapman, $s=k$ is an odd integer, which gives $2K(s,\tau )$.] □

Finally, the $s=0$ case again leads to the eta-transformation formula (Equation (32)) based on the following:

$$A(0,\tau )={\displaystyle \frac{\pi i\tau }{12}}-log\eta \left(\tau \right).$$

(51)

Proof of Theorem 3.

A proof is given by Kurokawa, but Lewittes could have proved it based on the following dissection:

$$F(s,\tau )=\zeta (s,\tau )+e^{-\pi is}\zeta (s,1-\tau ),$$

(52)

Hence,

$$F^{\prime }(0,\tau )={\zeta }^{\prime }(0,\tau )+{\zeta }^{\prime }(0,1-\tau )-\pi i\zeta (0,1-\tau ).$$

Substituting the Lerch formula ${\zeta }^{\prime }(0,\tau )=log{\displaystyle \frac{\mathsf{\Gamma }\left(\tau \right)}{\sqrt{2\pi }}}$ and $\zeta (0,\tau )=-B_1\left(\tau \right)$, we deduce that

$$\begin{array}{cc}\hfill F^{\prime }(0,\tau )& =log{\displaystyle \frac{\mathsf{\Gamma }\left(\tau \right)\mathsf{\Gamma }(1-\tau )}{2\pi }}-\pi i\left(\tau -{\displaystyle \frac{1}{2}}\right)\hfill \\ \hfill & =-log2sin\pi \tau -\pi i\left(\tau -{\displaystyle \frac{1}{2}}\right),\hfill \end{array}$$

based on the reciprocity relation. Hence,

$$\begin{array}{c}\hfill e^{-F^{\prime }(0,\tau )}=2sin\pi \tau e^{\pi i\left(\tau -{\displaystyle \frac{1}{2}}\right)},\end{array}$$

which leads to the first formula of Equation (33). In the case of $\mathrm{Im}\tau <0$, the dissection (Equation (52)) is changed into the following:

$$F(s,\tau )=\zeta (s,\tau )+e^{\pi is}\zeta (s,1-\tau ),$$

and the argument proceeds similarly.

Recall Equation (44) as

$$G(s,\tau )=(1+e^{\pi is})\left(\zeta \left(s\right)+{\displaystyle \frac{{(-2\pi i)}^s}{\mathsf{\Gamma }\left(s\right)}}A(s,\tau )\right).$$

We observe the ease of computation at $s=0$. For as long as ${\displaystyle \frac{1}{\mathsf{\Gamma }\left(s\right)}}$ survives, the terms are 0. Hence, the following is needed:

$${(-2\pi i)}^s{\left(\mathsf{\Gamma }{\left(s\right)}^{-1}\right)}^{\prime }A(s,\tau ){|}_{s=0}=-\psi \left(s\right)\mathsf{\Gamma }{\left(s\right)}^{-1}{|}_{s=0}A(0,\tau )=A(0,\tau ),$$

because of

$$\begin{array}{c}\hfill \psi \left(s\right)-\psi (1-s)=-{\displaystyle \frac{1}{s}}-2s\sum _{n=1}^{\infty }{\displaystyle \frac{1}{s^2-n^2}},s\notin \mathbb{Z},\end{array}$$

(53)

which is part of Equation (20) and ${\displaystyle \frac{1}{\mathsf{\Gamma }\left(s\right)}}={\displaystyle \frac{s}{\mathsf{\Gamma }(s+1)}}$. Hence, the first equality of Equation (35) follows, and the second equality follows from Equation (51). □

Note that $F(s,\tau )$ for $s=k\in \mathbb{N}$ is the kth derivative of the cotangent function. Also note that Equation (36) is very plausible since it contains the Dedekind eta-function and is an analogue of Equation (33). One can see how it will be messy to work at $s=1$, in which case we need to evaluate many terms, and use is made of Equation (34). Then connecting two results (based on the functional equation) would produce some interesting results.

4. Generalized Ramanujan Formula as per Komori et al. as the Modular Relation for the Barnes Multiple Zeta-Function

We assemble results on the Barnes multiple and double zeta-functions, which are basic in Lewittes’ and Kurokawa’s results, and we derive Theorem 5 from Theorem 6 using the data on the Barnes double zeta-function.

Suppose at least three of ${\omega }_1,\cdots ,{\omega }_r$ are linearly independent and that all ${\omega }_i$s and $\alpha$ lie on the same side of some straight line L going through the origin. Let $\mathsf{\Lambda }$ denote the semi-lattice consisting of linear forms $\mathsf{\Omega }={\sum }_{j=1}^r{m}_j{\omega }_j$ with $(m_1,\dots ,m_r)\in {(\mathbb{N}\cup \left\{0\right\})}^r$.

Then, the Barnes multiple zeta-function (or r-ple Hurwitz zeta-function) ${\zeta }_r(s,\alpha ,\mathit{\omega })$ of the complex variable s with parameter $\alpha$ and basis $\mathit{\omega }$ is defined as the Dirichlet series (Equation (25)). Cf. [26,27].

From [26], we have the following:

Lemma 3.

$${\zeta }_r(s,z,\mathit{\omega })=-{\displaystyle \frac{\mathsf{\Gamma }(1-s)}{2\pi i}}{\int }_C{\displaystyle \frac{e^{-zw}{(-w)}^{s-1}}{{\prod }_{j=1}^r\left(1-e^{-{\omega }_{j}w}\right)}}\mathrm{d}w,$$

(54)

where the contour C can be taken as the one from $+\infty$ to λ along the real axis, going along the circle around 0 of radius λ counterclockwise to λ, and then going back to $+\infty$, where $0<\lambda <min\left\{\left|{\displaystyle \frac{2\pi }{{\omega }_j}}\right|\right\}$.

This expression gives the analytic continuation of ${\zeta }_r$ to the whole complex plane with simple poles at $s=1,\cdots ,r$; in particular, ${\zeta }_r$ is holomorphic at $s=0$. Also, the integrals along the real axis cancel each other for $s=1-m$ for $m\in \mathbb{N}$, so that the value is given by the residue of the integrand at $w=0$, which is as follows:

$${\zeta }_r(1-m,z,\mathit{\omega })={\displaystyle \frac{{(-1)}^r{}_r{S}_m^{\prime }(z,\mathit{\omega })}{m}},$$

(55)

where

$${}_r{S}_m^{\prime }(z,\mathit{\omega })={\displaystyle \frac{{\left({}^{1}B{\omega }_1+\cdots +{}^{r}B{\omega }_r+z\right)}^{m+r-1}m!}{(m+r-1)!{\prod }_{k=1}^r{\omega }_j}},$$

(56)

under the convention that the jth power ${\left({}^{i}B\right)}^j$ of ${}^{i}B$ is $B_j$.

We define the Stirling modular form ${\rho }_r\left(\mathit{\omega }\right)$ by

$$-log{\rho }_r\left(\mathit{\omega }\right)=\underset{z\to 0}{lim}\left[{\displaystyle \frac{\partial }{\partial s}}{\zeta }_r(0,z,\mathit{\omega })+logz\right],$$

and the r-ple gamma function ${\mathsf{\Gamma }}_r(z,\mathit{\omega })$ by

$$log\left({\displaystyle \frac{{\mathsf{\Gamma }}_r(z,\mathit{\omega })}{{\rho }_r\left(\mathit{\omega }\right)}}\right)={\displaystyle \frac{\partial }{\partial s}}{\zeta }_r(0,z,\mathit{\omega }).$$

(57)

We state the special case of Lemma 3.

Lemma 4.

We have the formula

$$\mathsf{\Gamma }\left(s\right)F(s,\tau )={\int }_0^{\infty }{\displaystyle \frac{u^s}{(e^u-1)(e^{\tau u}-1)}}{\displaystyle \frac{\mathrm{d}u}{u}},$$

(58)

for $\sigma >2$, $\mathrm{Im}\tau >0$, $\mathrm{Re}\tau >0$ and the formula for the double zeta-function:

$$\mathsf{\Gamma }\left(s\right){\zeta }_2(s,z,(1,\tau ))={\int }_0^{\infty }{\displaystyle \frac{u^s{e}^{(1+\tau -z)u}}{(e^u-1)(e^{\tau u}-1)}}{\displaystyle \frac{\mathrm{d}u}{u}}.$$

(59)

(54) amounts to

$${\zeta }_2(s,z,(1,\tau ))=-{\displaystyle \frac{\mathsf{\Gamma }(1-s)}{2\pi i}}{\int }_C{\displaystyle \frac{e^{-zw}{(-w)}^{s-1}}{\left(1-e^{-w}\right)\left(1-e^{-\tau w}\right)}}\mathrm{d}w,$$

or equivalently,

$$\mathsf{\Gamma }\left(s\right){\zeta }_2(s,z,(1,\tau ))={\displaystyle \frac{1}{e^{2\pi is}-1}}{\int }_C{\displaystyle \frac{w^{s-1}{e}^{(1+\tau -z)w}}{(e^w-1)(e^{\tau w}-1)}}\mathrm{d}w.$$

(60)

Proof. 

We have

$$\mathsf{\Gamma }\left(s\right)F(s,\tau )=\sum _{m,n=1}^{\infty }{\int }_0^{\infty }{u}^s{e}^{-mu}{e}^{-n\tau u}{\displaystyle \frac{\mathrm{d}u}{u}}={\int }_0^{\infty }{\displaystyle \frac{u^s{e}^{-u}{e}^{-\tau u}}{(1-e^{-u})(1-e^{-\tau u})}}{\displaystyle \frac{\mathrm{d}u}{u}},$$

upon summing the geometric series. Hence, multiplying the integrand by ${\displaystyle \frac{e^u{e}^{\tau u}}{e^u{e}^{\tau u}}}$, we deduce Equation (58).

In the case of the double zeta-function, we have the following:

$$\begin{array}{cc}\hfill \mathsf{\Gamma }\left(s\right){\zeta }_2(s,z,(1,\tau ))& =\sum _{m,n=0}^{\infty }{\int }_0^{\infty }{u}^s{e}^{-zu}{e}^{-mu}{e}^{-n\tau u}{\displaystyle \frac{\mathrm{d}u}{u}}\hfill \\ \hfill & ={\int }_0^{\infty }{\displaystyle \frac{u^s{e}^{-zu}}{(1-e^{-u})(1-e^{-\tau u})}}{\displaystyle \frac{\mathrm{d}u}{u}},\hfill \end{array}$$

We obtain Equation (59) in the same way.

It turns out that the only difference is whether one starts summing the geometric series from 1 or 0, which gives the difference to the effect that there appears the additional factor $e^{(1+\tau -z)u}$ (in the general case $e^{({\omega }_1+\cdots +{\omega }_r-z)u}$) or not. After this, it is a well-known Hankel contour method with the specified contour $C_{\lambda }$, cf. e.g., [28], §12.22. □

Let $H\left(\theta \right)$ be the open half-plane whose normal vector is $e^{i\theta }$ ([26], p. 388), and let

$$a=a\left(\mathit{y}\right)=a(y_1,\cdots ,y_r)={\omega }_1(1-y_1)+\cdots +{\omega }_1(1-y_r)\in H\left(\theta \right).$$

(61)

We write $a\left(y\right)=a(y,\cdots ,y)$.

Theorem 6

([9], Theorem 2.1). The functional equation holds true.

$$\begin{array}{cc}\hfill & {\zeta }_r(s,a\left(y\right),\mathit{\omega })\hfill \\ \hfill & =-{\displaystyle \frac{2\pi i}{\mathsf{\Gamma }\left(s\right)(e^{2\pi is}-1)}}\sum _{k=1}^r\sum _{m\in \mathbb{Z}\setminus \left\{0\right\}}{\omega }_k^{-1}\prod _{\begin{array}{c}j=1\\ j\ne k\end{array}}^r\left({\displaystyle \frac{e^{(2\pi im{\omega }_j/{\omega }_k)y}}{e^{2\pi im{\omega }_j/{\omega }_k}-1}}\right){\left({\displaystyle \frac{{\omega }_k}{2\pi im}}\right)}^{1-s}{e}^{2\pi imy},\hfill \end{array}$$

(62)

where $\theta -\pi arg{\displaystyle \frac{{\omega }_k}{2\pi im}}<\theta$ and the right-hand side converges absolutely and uniformly in the whole plane $\mathbb{C}$ if $0<y<1$ and in the region $\sigma <0$ if $y=0$.

As is noted in [29], p. 3 and [9], pp. 55–56, the proof of Theorem 6 follows the proof in [30]. Matsumoto [29] established a proof of the functional equation for the double zeta-function

$${\tilde{\zeta }}_2(u,v,z,\omega )=\sum _{m=0}^{\infty }{(z+m)}^{-u}\sum _{n=0}^{\infty }{(z+m+n\tau )}^{-v},{\zeta }_2(v,z,\omega )={\tilde{\zeta }}_2(0,v,z,\omega ),$$

for $z>0,\omega >0$. In [31], an asymptotic formula for the special case ${\zeta }_2^{\prime }(v,z,\omega )$ is established.

For each $k\in \{1,\cdots ,n\}$, let $I_k^{+}=\{j\in \{1,\cdots ,n\}\setminus \left\{k\right\}|arg({\omega }_j/{\omega }_k)>0\}$, and similarly, we define $I_k^{-}$. For $K\subset \{1,\cdots ,n\}$, we define

$$\delta \left(J\right)=\left\{\begin{array}{cc}0\hfill & ifJ\ne \varnothing \hfill \\ {(-1)}^{r+1}\hfill & ifJ=\varnothing .\hfill \end{array}\right.$$

Theorem 6 for the case of the generalized Eisenstein series ((25) with $z=a\left(0\right)$; cf. the special case Equation (45)), reads as follows:

Corollary 4

([9], Corollary 2.2). We have

$$\begin{array}{cc}\hfill & {\zeta }_r(s,a\left(0\right),\mathit{\omega })=-{\displaystyle \frac{2\pi i}{\mathsf{\Gamma }\left(s\right)(e^{2\pi is}-1)}}\sum _{k=1}^r{\omega }_k^{-1}\hfill \\ \hfill & \times \left(S_k^{-}(s,\mathit{\omega })+S_k^{+}(s,\mathit{\omega })+\delta \left(I_k^{-}\right){\left({\displaystyle \frac{{\omega }_k}{2\pi i}}\right)}^{1-s}\zeta (1-s)+\delta \left(I_k^{+}\right){\left(-{\displaystyle \frac{{\omega }_k}{2\pi i}}\right)}^{1-s}\zeta (1-s)\right),\hfill \end{array}$$

where the series on the right-hand side converge absolutely and uniformly in $\mathbb{Z}$ and where

$$\begin{array}{cc}\hfill & S_k^{-}(s,\mathit{\omega })=\sum _{m=1}^{\infty }\prod _{\begin{array}{c}j=1\\ j\ne k\end{array}}^r\left({\displaystyle \frac{1}{e^{2\pi im{\omega }_j/{\omega }_k}-1}}-\delta \left(I_k^{-}\right)\right){\left({\displaystyle \frac{{\omega }_k}{2\pi im}}\right)}^{1-s}\hfill \\ \hfill & S_k^{+}(s,\mathit{\omega })=\sum _{m=1}^{\infty }\prod _{\begin{array}{c}j=1\\ j\ne k\end{array}}^r\left({\displaystyle \frac{1}{e^{-2\pi im{\omega }_j/{\omega }_k}-1}}-\delta \left(I_k^{+}\right)\right){\left(-{\displaystyle \frac{{\omega }_k}{2\pi im}}\right)}^{1-s}.\hfill \end{array}$$

To find special values, we appeal to Equation (55). Therefore, we recall the nth Bernoulli polynomial $B_n\left(x\right)$ and denote its generating function by $f(z,x)={\displaystyle \frac{z{e}^{xz}}{e^z-1}}$.

Lemma 5.

As $z\to k\in \mathbb{Z}$, we have

$$\begin{array}{c}\hfill {\displaystyle \frac{\mathsf{\Gamma }\left(s\right)(e^{2\pi is}-1)}{2\pi i}}=\left\{\begin{array}{cc}{\displaystyle \frac{{(-1)}^k}{(-k)!}}+O(s-k)\hfill & k\le 0\hfill \\ (k-1)!(s-k)+O\left({(s-k)}^2\right)\hfill & k>0\hfill \end{array}\right..\end{array}$$

From this, we have for $l\in \mathbb{Z}$ or $l>r$ if $y=0$

$$\begin{array}{c}\hfill {\displaystyle \frac{\mathsf{\Gamma }\left(s\right)(e^{2\pi is}-1)}{2\pi i}}{\zeta }_r(s,a\left(y\right),\mathit{\omega }){|}_{s=r-l}=\sum _{\begin{array}{c}{m}_1,\cdots ,m_r\ge 0\\ m_1+\cdots +m_r=l\end{array}}\prod _{j=1}^r{\displaystyle \frac{B_{m_j}\left(y\right)}{m_j!}}{\omega }_j^{m_j-1}.\end{array}$$

In particular,

$$\begin{array}{c}\hfill {\displaystyle \frac{\mathsf{\Gamma }\left(s\right)(e^{2\pi is}-1)}{2\pi i}}{\zeta }_2(s,a\left(0\right),(1,\tau ){|}_{s=2-l}=\sum _{k=0}^l{\displaystyle \frac{B_k{B}_{l-k}}{k!(l-k)!}}{\tau }^{k-1}.\end{array}$$

(63)

Lemma 6.

We have the closed-form evaluation:

$$\begin{array}{cc}\hfill & -\sum _{m=1}^{\infty }{\left({\displaystyle \frac{1}{2\pi im}}\right)}^{1-s}{\displaystyle \frac{2e^{2\pi im\tau }}{1-e^{2\pi im\tau }}}+{\tau }^{-1}\sum _{m=1}^{\infty }{\left({\displaystyle \frac{\tau }{2\pi im}}\right)}^{1-s}{\displaystyle \frac{2e^{-2\pi im{\tau }^{-1}}}{1-e^{-2\pi im{\tau }^{-1}}}}\hfill \\ \hfill & =-\sum _{k=0}^{2-s}{\displaystyle \frac{B_k{B}_{2-s-k}}{k!(2-s-k)!}}{\tau }^{k-1}\hfill \\ \hfill & +\left\{\begin{array}{cc}{\left({\displaystyle \frac{1}{2\pi i}}\right)}^{1-s}\left(1-{\tau }^{-s}\right)\zeta (1-s)\hfill & s\ne 0\hfill \\ {\displaystyle \frac{1}{2\pi i}}log{\displaystyle \frac{\tau }{i}}\hfill & s=0\hfill \end{array}\right.,\hfill \end{array}$$

(64)

for $1-s$ an odd integer. This gives the closed-form evaluation of $K(s,\tau )$ in Equation (45) in view of Equation (45).

Proof. 

We apply Corollary 4 with ${\omega }_1=1$, ${\omega }_2=\tau$. Then, $arg\left({\displaystyle \frac{{\omega }_1}{{\omega }_2}}\right)=arg{\tau }^{-1}<0$ and $arg\left({\displaystyle \frac{{\omega }_2}{{\omega }_1}}\right)=arg\tau >0$. Hence, we have the following

Table 1. Values of $\delta \left(I_k^{\pm }\right)$.

$\mathit{k}=$	$\mathit{\delta }\left({\mathit{I}}_{\mathit{k}}^{\mathbf{+}}\right)$	$\mathit{\delta }\left({\mathit{I}}_{\mathit{k}}^{\mathbf{-}}\right)$
1	0	$-1$
2	$-1$	0

:

Incorporating these data in the equalities in Corollary 4, we deduce that

$$\begin{array}{cc}\hfill & -{\displaystyle \frac{\mathsf{\Gamma }\left(s\right)(e^{2\pi is}-1)}{2\pi i}}{\zeta }_2(s,a\left(0\right),(1,\tau ))\hfill \\ \hfill & +{\left({\displaystyle \frac{1}{2\pi i}}\right)}^{1-s}\zeta (1-s)+{\tau }^{-1}{\left(-{\displaystyle \frac{\tau }{2\pi i}}\right)}^{1-s}\zeta (1-s)\hfill \\ \hfill & =\sum _{m=1}^{\infty }\left({\displaystyle \frac{e^{2\pi im\tau }}{1-e^{2\pi im\tau }}}\right)\left({\left(-{\displaystyle \frac{1}{2\pi im}}\right)}^{1-s}-{\left({\displaystyle \frac{1}{2\pi im}}\right)}^{1-s}\right)\hfill \\ \hfill & +{\tau }^{-1}\sum _{m=1}^{\infty }\left({\displaystyle \frac{e^{-2\pi im{\tau }^{-1}}}{1-e^{-2\pi im{\tau }^{-1}}}}\right)\left({\left({\displaystyle \frac{\tau }{2\pi im}}\right)}^{1-s}-{\left(-{\displaystyle \frac{\tau }{2\pi im}}\right)}^{1-s}\right).\hfill \end{array}$$

(65)

We immediately observe that if $1-s$ is an even integer, Equation (65) leads to the well-known relation between the zeta-values and Bernoulli numbers, while if $1-s$ is an odd integer, Equation (65) amounts to the following:

$$\begin{array}{cc}\hfill & -\sum _{m=1}^{\infty }{\left({\displaystyle \frac{1}{2\pi im}}\right)}^{1-s}{\displaystyle \frac{2e^{2\pi im\tau }}{1-e^{2\pi im\tau }}}+{\tau }^{-1}\sum _{m=1}^{\infty }{\left({\displaystyle \frac{\tau }{2\pi im}}\right)}^{1-s}{\displaystyle \frac{2e^{-2\pi im{\tau }^{-1}}}{1-e^{-2\pi im{\tau }^{-1}}}}\hfill \\ \hfill & =-{\displaystyle \frac{\mathsf{\Gamma }\left(s\right)(e^{2\pi is}-1)}{2\pi i}}{\zeta }_2(s,a\left(0\right),(1,\tau ))\hfill \\ \hfill & +{\displaystyle \frac{1}{2\pi i}}\left({\left({\displaystyle \frac{1}{2\pi i}}\right)}^{-s}-{\left(-{\displaystyle \frac{\tau }{2\pi i}}\right)}^{-s}\right)\zeta (1-s),\hfill \end{array}$$

(66)

where the right-hand side is the value at $1-s$, being an odd integer including the case $s=0$. Indeed, to include the case, we have to consider the limit case as $s\to 0$ of the last term $L\left(s\right)$, say, of Equation (66).

We find that as $s\to 0$

$$\begin{array}{c}\hfill L\left(s\right)={\displaystyle \frac{1}{2\pi i}}\left(-log{\displaystyle \frac{1}{2\pi i}}+log{\displaystyle \frac{-\tau }{2\pi i}}\right)+o\left(1\right)={\displaystyle \frac{1}{2\pi i}}\left(log{\displaystyle \frac{\tau }{i}}-{\displaystyle \frac{\pi }{2}}i\right)+o\left(1\right),\end{array}$$

(67)

where $log{\displaystyle \frac{1}{i}}=-{\displaystyle \frac{\pi }{2}}i$, so that

$$\left|arg{\displaystyle \frac{\tau }{i}}\right|<{\displaystyle \frac{\pi }{2}}.$$

(68)

Cf. [17], §69. We understand Equation (67) to mean the following:

$$L\left(s\right)={\displaystyle \frac{1}{2\pi i}}log{\displaystyle \frac{\tau }{i}}+o\left(1\right),s\to 0,$$

(69)

under Equation (68).

Incorporating Equations (69) and (63), we transform Equation (66) into Equation (64).

□

Proof of Theorem 5.

Proof of Theorem 5 follows from Lemma 6 as follows. The $\varkappa =0$ case of Equation (79) leads to Equation (32) as in Corollary 3.

It suffices to derive Equation (70) from Equation (62).

Using

$${\left({\displaystyle \frac{1}{\mathsf{\Gamma }\left(s\right)(e^{2\pi is}-1)}}\right)}^{\prime }{|}_{s=0}=2\pi i(\gamma +2\pi i),$$

we find from Equation (62) that

$$\begin{array}{cc}\hfill & {\zeta }_r^{\prime }(0,a(1/2),\mathit{\omega })\hfill \\ \hfill & =(\gamma +2\pi i)\sum _{k=1}^r\sum _{m\in \mathbb{Z}\setminus \left\{0\right\}}{\displaystyle \frac{{(-1)}^m}{m}}\prod _{\begin{array}{c}j=1\\ j\ne k\end{array}}^r\left({\displaystyle \frac{e^{\pi im\tau }}{1-e^{2\pi im\tau }}}\right)\hfill \\ \hfill & -\sum _{k=1}^r\sum _{m\in \mathbb{Z}\setminus \left\{0\right\}}{(-1)}^m\prod _{\begin{array}{c}j=1\\ j\ne k\end{array}}^r\left({\displaystyle \frac{e^{\pi im{\omega }_j/{\omega }_k}}{e^{2\pi im{\omega }_j/{\omega }_k}-1}}\right){\displaystyle \frac{1}{2\pi im}}log{\displaystyle \frac{{\omega }_k}{2\pi im}}.\hfill \end{array}$$

(70)

The coefficient ${\displaystyle \frac{e^{\pi im{\omega }_j/{\omega }_k}}{e^{2\pi im{\omega }_j/{\omega }_k}-1}}$ may be expressed as ${\displaystyle \frac{1}{sinh2\pi im{\omega }_j/{\omega }_k}}$ and also as $-{\sum }_{n=0}^{\infty }{e}^{\pi i(2m+1)n{\omega }_j/{\omega }_k}$. Hence, (39) follows as a special case of Equation (70). □

5. Kronecker Limit Formula Without Absolute Value as per Kurokawa

Kurokawa [10] considered the Dirichlet series:

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

(71)

absolutely convergent for $\sigma >2$ under the condition $0<\mathrm{Im}z<\mathrm{Im}\tau$ with the branch such that

$$-\pi <arglog(m+n\tau +z)<\pi .$$

(72)

The presence of z with the condition $0<\mathrm{Im}z$ makes the difference between Kurokawa’s work and other works of Lewittes et al.

Kurokawa divides the sum into four parts: (i)${}_{K}m$$\ge 0,n\ge 0$, (ii)${}_{K}m$$<0,n<0$, (iii)${}_{K}m$$<0,n\ge 0$, and (iv)${}_{K}m$$\ge 0,n<0$ (see

Table 2. Kurokawa dissection.

summand	$m\ge 0,n\ge 0$	$m<0,n<0$
term	$m+n\tau +z$	$e^{-\pi i}(-m-1+(-n-1)\tau +1+\tau -z)$
Zeta	${\zeta }_2(s,z,(1,\tau ))$	$e^{\pi is}{\zeta }_2(s,1+\tau -z,(1,\tau ))$
summand	$m\ge 0,n<0$	$m<0,n\ge 0$
term	$m+(-n-1)(-\tau )+z-\tau$	$e^{\pi i}(-m-1+n(-\tau )+1-z)$
Zeta	${\zeta }_2(s,z-\tau ,(1,-\tau ))$	$e^{-\pi is}{\zeta }_2(s,1-z,(1,-\tau ))$

). Corresponding to the last three cases, we express $m+n\tau +z$ as $e^{-\pi i}(-m-1+(-n-1)\tau +1+\tau -z)$, $e^{\pi i}(-m-1+n(-\tau )+1-z)$, and $m+(-n-1)(-\tau )+z-\tau$, so that the arguments remain in the range of Equation (72). Hence, we obtain

$$\begin{array}{cc}\hfill \phi (s,\tau ,z)& ={\zeta }_2(s,z,(1,\tau ))+e^{\pi is}{\zeta }_2(s,1+\tau -z,(1,\tau ))\hfill \\ \hfill & +e^{-\pi is}{\zeta }_2(s,1-z,(1,-\tau ))+{\zeta }_2(s,z-\tau ,(1,-\tau )).\hfill \end{array}$$

(73)

Theorem 7

([10]). We have the Kronecker limit formula without absolute value:

$$\prod _{m,n=-\infty }^{\infty }(m+n\tau +z)=(1-q_z)\prod _{n=-\infty }^{\infty }(1-q_{\tau }^n{q}_z)(1-q_{\tau }^n{q}_z^{-1}).$$

(74)

Proof. 

Through zeta-regularization, we are to transform $e^{-{\zeta }^{\prime }(0,z,\tau )}$ into the right-hand side of Equation (74).

From Equation (73), we see that ${\phi }^{\prime }(0,\tau ,z)$ is the sum of derivatives of four zeta-functions at $s=0$ plus $-\pi i({\zeta }_2(0,1-z,(1,-\tau ))-{\zeta }_2(0,1+\tau -z,(1,\tau ))=-\pi iM(z,\tau )$, say.

First, we deal with $M(z,\tau )$. Based on Equation (75), we have

$${\zeta }_2(0,1-z,(1,-\tau ))=-{\zeta }_2(0,1+\tau -z,(1,\tau ))]=-{\displaystyle \frac{1}{\tau }}\left(B_2\left(z\right)+{\displaystyle \frac{{\tau }^2}{6}}-\tau z+{\displaystyle \frac{\tau }{2}}\right),$$

Hence,

$$M(z,\tau )=-{\displaystyle \frac{1}{\tau }}\left(B_2\left(z\right)+{\displaystyle \frac{{\tau }^2}{6}}-\tau z+{\displaystyle \frac{\tau }{2}}\right).$$

Recall Equations (55) and (56) in the case of $r=2$ and $m=1$.

$${\zeta }_2(0,z,{\omega }_1,{\omega }_2)={}_2{S}_1\prime (z,{\omega }_1,{\omega }_2),$$

where

$${}_2{S}_1\prime (z,{\omega }_1,{\omega }_2)={\displaystyle \frac{{\left({}^{1}B{\omega }_1+{}^{1}B{\omega }_2+z\right)}^2}{2{\omega }_1{\omega }_2}}.$$

Hence,

$${\zeta }_2(0,z,({\omega }_1,{\omega }_2))={\displaystyle \frac{1}{2{\omega }_1{\omega }_2}}\left(B_2{\omega }_1^2+B_2{\omega }_2^2+z^2+2(B_1^2{\omega }_1{\omega }_2+B_{1}z{\omega }_1+B_{1}z{\omega }_2)\right).$$

(75)

Now, we treat the product of derivatives. To simplify the calculation, we use the Kurokawa r-ple gamma function ${\mathsf{\Gamma }}_r^K(z,\mathit{\omega })$ not based on Equation (57) but based on the following:

$${\mathsf{\Gamma }}_r^K{(z,\mathit{\omega })}^{-1}=e^{-{\zeta }_r^{\prime }(0,z,\mathit{\omega })}=\prod _{m_1,\cdots ,m_r=0}^{\infty }(z+m_1{\omega }_1+\cdots +m_r{\omega }_r).$$

Then, Equation (57) reads as follows:

$${\zeta }_2^{\prime }(0,z,(1,{\omega }_2))=log\left({\mathsf{\Gamma }}_2^K(z,(1,{\omega }_2)\right).$$

Hence,

$$\begin{array}{c}\hfill e^{-{\zeta }_2^{\prime }(0,z,(1,\tau ))}={\mathsf{\Pi }}_S{e}^{\pi iM(z,\tau )},\end{array}$$

(76)

where

$$\begin{array}{cc}\hfill & {\mathsf{\Pi }}_S={\mathsf{\Gamma }}_2^K{(z,(1,\tau ))}^{-1}{\mathsf{\Gamma }}_2^K{(1-z,(1,-\tau ))}^{-1}{\mathsf{\Gamma }}_2^K{(1+\tau -z,(1,\tau ))}^{-1}{\mathsf{\Gamma }}_2^K{(z-\tau ,(1,-\tau ))}^{-1}\hfill \\ \hfill & =2q_{\tau }^{{\displaystyle \frac{1}{12}}}sin\pi z{e}^{{\displaystyle \frac{\pi i}{\tau }}{B}_2\left(z\right)}\prod _{n=-\infty }^{\infty }(1-q_{\tau }^n{q}_z)(1-q_{\tau }^n{q}_z^{-1}).\hfill \end{array}$$

[32], Proposition 6. Substituting this into Equation (76), and noting that

$$2q_{\tau }^{{\displaystyle \frac{1}{12}}}sin\pi z{e}^{{\displaystyle \frac{\pi i}{\tau }}{B}_2\left(z\right)}{e}^{\pi iM(z,\tau )}=1-q_z,$$

we conclude Equation (74). □

6. Appendix: Some Notes on Ramanujan’s Formula as the Modular Relation

In [33], pp. 275–276, Entry 21 (i), Chapter 14, it is claimed that Entry 21 (i) yields the $k<0$ case in [33], p. 261, Entry 13, so that the generalized Ramanujan formula for all integers but 0 is in order upon modifying the residual function $\mathrm{P}\left(x\right)$, which is ${\alpha }^{-k}{\sum }_{n=1}^{\infty }{\displaystyle \frac{n^{-2k-1}}{e^{2\alpha n}-1}}+{(-1)}^{k+1}{\beta }^{-k}{\sum }_{n=1}^{\infty }{\displaystyle \frac{n^{-2k-1}}{e^{2\beta n}-1}}$ of Equation (24). But, this is not convenient, and we stated explicit formulas above for the residual function in the form of the Bochner modular relation (or its upper half-plane verison—the Hecke correspondence).

By changing the variable

$$x=-i\tau ,$$

(77)

we have the correspondence: the right half-plane $\mathcal{RHP}=\{x|\mathrm{Re}x>0\}$↔ the upper half-plane $\mathcal{H}$.

Based on Equation (77), Equation (13) becomes the upper half-plane version

$$\sum _{m=1}^{\infty }{\displaystyle \frac{a_m}{e^{-im\tau }-1}}=\sum _{n=1}^{\infty }{b}_n{e}^{in\tau },\mathrm{Im}\tau >0,$$

where the right-hand side is more often recognized as a Fourier series (or a q-expansion, or a Laurent expansion, etc.).

Problem: Is it possible to draw some information from the functional equation when $\alpha$ is not necessarily an integer?

Even in the case of $\alpha$ being an even integer, (5) leads to the following:

$$\begin{array}{cc}\hfill & \mathsf{\Gamma }(1-s)\phi (1-s)\hfill \\ \hfill & =\sum _{j=0}^2\left({\displaystyle \genfrac{}{}{0pt}{}{2}{j}}\right)e^{-\pi i(1-j)s}\left({\left(2\pi \right)}^{-2s+\alpha }{(-1)}^{{\displaystyle \frac{\alpha }{2}}}\mathsf{\Gamma }{(s-\alpha )}^2\mathsf{\Gamma }(\alpha +1-s)\right)\phi (s-\alpha ),\hfill \end{array}$$

(78)

where the expansion holds for $\sigma >\alpha +1$.

Bellman’s work [34] applies in place of Equation (78)

$$\begin{array}{c}\hfill \mathsf{\Gamma }(1-s)\phi (1-s)=\left({\left(2\pi \right)}^{-2s+\alpha +1}{(-1)}^{{\displaystyle \frac{\alpha }{2}}}\left(cot{\displaystyle \frac{\pi }{2}}s\right)\mathsf{\Gamma }(s-\alpha )\right)\phi (s-\alpha ).\end{array}$$

Both are rather involved. Cf. [35].

After changing into the rapidly convergent series, it is customary to use Equation (15) and express Equation (24) as the relation in Equation (14) between the correspondence $x\leftrightarrow {\displaystyle \frac{1}{x}}$ in $x\in \mathcal{RHP}$.

The upper half-plane version of Equation (14) ([36], (2.4), (2.5)) reads as follows, with $\tau \in \mathcal{H}$ and any integer $\varkappa \ge 0$:

$$\begin{array}{cc}\hfill & \sum _{k=1}^{\infty }{\sigma }_{-2\varkappa -1}\left(k\right)e^{2\pi ik\tau }-{\tau }^{2\varkappa }\sum _{k=1}^{\infty }{\sigma }_{-2\varkappa -1}\left(k\right)e^{-{\displaystyle \frac{2\pi ik}{\tau }}}={\mathrm{P}}_{2\varkappa +1}(-2\pi i\tau ),\hfill \\ \hfill & {\mathrm{P}}_{2\varkappa +1}(-2\pi i\tau )={\displaystyle \frac{{(-1)}^{\varkappa +1}{\left(2\pi \right)}^{2\varkappa +1}}{2i\tau }}\sum _{j=0}^{\varkappa +1}{(-1)}^j{\displaystyle \frac{B_{2j}}{\left(2j\right)!}}{\displaystyle \frac{B_{2\varkappa +2-2j}}{(2\varkappa +2-2j)!}}{\tau }^{2\varkappa +2-2j}\hfill \\ \hfill & +\left\{\begin{array}{cc}-{\displaystyle \frac{1}{2}}\zeta (2\varkappa +1)\left\{1-{\tau }^{2\varkappa }\right\}\hfill & if\varkappa \ge 1,\hfill \\ {\displaystyle \frac{1}{2}}log{\displaystyle \frac{\tau }{i}}\hfill & if\varkappa =0,\hfill \end{array}\right.,\hfill \end{array}$$

(79)

where $\left|arg{\displaystyle \frac{\tau }{i}}\right|<{\displaystyle \frac{\pi }{2}}$. In this form, it is an example of the Hecke modular relation and may be referred to as a q-expansion. In general, for a modular form f, the Fourier series

$$f\left(\tau \right)=\sum _{n=0}^{\infty }{a}_n{q}^n=\sum _{n=0}^{\infty }{a}_n{e}^{2\pi in\tau },$$

(80)

is often called the q-expansion.

Note that we changed the notation $k>0$ in Equation (14) to $\varkappa \ge 0$ to include the case of $\varkappa =0$. This inclusion is important, since the $\varkappa =0$ case implies the eta-transformation formula (Equation (32)).

Generalization of Equation (24) ($k>0$) is performed in [37], pp. 429–432, Entry 20 and [33], pp. 253–254, Entry 8. Let $\alpha ,\beta ,t>0$ with $\alpha \beta =\pi$ and $\alpha /\beta =t$ and let $\phi \left(z\right)$ be an entire function. Then, with $f\left(z\right)={\displaystyle \frac{\phi \left(2\beta Mz\right)}{z^{n+1}\left(e^{2\pi iMz/t}-1\right)}}$, ${\displaystyle \frac{g^{\prime }}{g}}\left(z\right)={\displaystyle \frac{1}{e^{-2\pi Mz}-1}}$ and C the rectangle with vertices at $\pm i,\pm t$, the following lemma corresponds to Entry 20, where $M=m+{\displaystyle \frac{1}{2}}$ and m is a positive integer.

Lemma 7.

The following generalization of the argument principle holds:

$${\displaystyle \frac{1}{2\pi i}}{\int }_{C}f\left(z\right){\displaystyle \frac{g^{\prime }}{g}}\left(z\right)\mathrm{d}z=\sum _{j=1}^l{n}_{j}g\left({\alpha }_j\right)-\sum _{i=1}^k{p}_{i}g\left(z_i\right)+\sum _{n=1}^p\underset{z={\zeta }_n}{\mathrm{Res}}f\left(z\right){\displaystyle \frac{g^{\prime }}{g}}\left(z\right),$$

if $g\left(z\right)$ has finitely many poles $z_i$, $1\le i\le k$ with multiplicity $p_i$ resp. finitely many zeros ${\alpha }_j$, $1\le j\le l$ with multiplicity $n_j$, and $f\left(z\right)$ is meromorphic with poles at ${\zeta }_n$, $1\le n\le p$ (assuming that poles of f and ${\displaystyle \frac{g^{\prime }}{g}}$ do not coincide for simplicity) in a domain D with boundary C.

In view of the global expressions

$$\begin{array}{c}\hfill \pi cot\pi z={\displaystyle \frac{1}{z}}+2z\sum _{n=1}^{\infty }{\displaystyle \frac{1}{z^2-n^2}}=-\pi i-{\displaystyle \frac{2\pi i}{e^{-2\pi iz}-1}}=\pi i+{\displaystyle \frac{2\pi i}{e^{2\pi iz}-1}},z\notin \mathbb{Z},\end{array}$$

(81)

$g\left(z\right)$ in Lemma 7 can be chosen to be the cotangent function $\pi cot\pi z$, then Entry 20 is a consequence of Lemma 7. In view of the fact that the partial fraction expansion for the cotangent function is equivalent to the functional equation for the Riemann zeta-function and that the cotangent function is essentially the polylogarithm of order 0, it is the most natural way to derive Ramanujan’s formula and the eta transformation formula, cf. [38,39,40].

In the above, we were restricted to the case of the Hecke modular relation or the Hecke correspondence, which is equivalent to the functional equation with the single gamma factor $\mathsf{\Gamma }\left(s\right)$. More generally, this is called the RHB (Riemann–Hecke–Bochner) correspondence. We shall mention generalities of the Hecke correspondence, cf. [6,41].

In relation to the q-expansion (Equation (80)), we introduce the modular-type functions corresponding to the Dirichlet series (Equation (83)):

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

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

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

(82)

Definition 3.

Let

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

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

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

(83)

which we assume are absolutely convergent for $\sigma >{\sigma }_a^{\ast }$ and $\sigma >{\sigma }_b^{\ast }$, respectively. Then, $\phi \left(s\right)$ and $\psi \left(s\right)$ are said to satisfy Hecke’s functional equation (HFE)

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

(84)

where $A>0$ is a constant if there exists a regular function $\chi \left(s\right)$ outside of a compact set $\mathcal{S}$ such that

$$\chi \left(s\right)=A^{-s}\mathsf{\Gamma }\left(s\right)\phi \left(s\right),\sigma >\alpha (\ge {\sigma }_a^{\ast })$$

and

$$\chi \left(s\right)=A^{-(r-s)}\mathsf{\Gamma }(r-s)\psi (r-s),\sigma <\beta (\le r-{\sigma }_b^{\ast })$$

and such that $\chi \left(s\right)$ is convex, in the sense that

$$e^{-\epsilon |t|}\chi (\sigma +it)=O\left(1\right),0<\epsilon <{\displaystyle \frac{\pi }{2}},$$

uniformly in $\sigma ,{\sigma }_1\le \sigma \le {\sigma }_2,|t|\to \infty$. We refer to the Dirichet series (Equation (83)) as Hecke L-functions (HLF) which satisfy (HFE) (Equation (84)).

Following Bochner [42], the residual function is defined as follows:

$$\mathrm{P}\left(x\right)={\displaystyle \frac{1}{2\pi i}}{\int }_C\chi \left(s\right)x^{-s}ds,$$

where C encircles all the singularities of $\chi \left(s\right)$ in $\mathcal{S}$.

Then, Equation (84) needs to become Equation (85), being modified by replacing $\psi (r-s)$ with $C\psi (r-s)={\sum }_{n=1}^{\infty }{\displaystyle \frac{C{b}_n}{{\mu }_n^{r-s}}}$.

It is known that the partial fraction expansion (Equation (81)) is equivalent to the functional equation for the Riemann zeta-function, and in the long run, it is equivalent to the above modular relations.

Lemma 8

(Hecke). The Dirichlet series (Equation (83)) satisfies the following functional equation:

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

(85)

and $A^{-s}\mathsf{\Gamma }\left(s\right)\phi \left(s\right)+{\displaystyle \frac{a_0}{s}}+{\displaystyle \frac{C{b}_0}{r-s}}$ is BEV (bounded in every vertical strip) and equivalent to the modular transformation (Equation (82)).

Lemma 8 is a useful statement of Hecke’s epoch-making discovery [43,44].

7. Conclusions

We discussed the papers of Chapman, Komori et al., Kurokawa, and Lewittes from the point of view of the lattice (zeta-)function (through dissection that is intrinsically connected with it). We deduced (generalizations of) Ramanujan’s formula and evaluated the derivative of relevant zeta-functions at $s=0$, making clear that all of them are consequences of modular relations. Thus, we arrive at the Kronekcer limit formula in Stark’s sense and derive determinants associated with strings.