The Cotangent Function as an Avatar of the Polylogarithm Function of Order 0 and Ramanujan’s Formula

Abstract

In this paper we will be concerned with zeta-symmetry—the functional equation for the (Riemann) zeta-function (equivalents to which are called modular relations)—and reveal the reason why so many results are intrinsic to PFE (Partial Fraction Expansion) for the cotangent function. The hidden reason is that the cotangent function (as a function in the upper half-plane, say) is the polylogarithm function of order 0 (with complex exponential argument), and therefore it shares properties intrinsic to the Lerch zeta-function of order 0. Here we view the Lerch zeta-function defined in the unit circle as a zeta-function in a wider sense, as a function defined in the upper and lower half-planes. As evidence, we give a plausibly most natural proof of Ramanujan’s formula, including the eta transformation formula as a consequence of the modular relation via the cotangent function, speculating the reason why Ramanujan had been led to such a formula. Other evidence includes the pre-Poisson summation formula as the pick-up principle (which in turn is a generalization of the argument principle).

1. Introduction

Ramanujan’s formula in its original form reads for every integer $\varkappa \ge 1$ and $\alpha >0$, $\beta >0$ satisfying the relation

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

as a modular relation for Lambert series

$$\begin{array}{cc}\hfill & {\alpha }^{-\varkappa }\left\{{\displaystyle \frac{1}{2}}\zeta (2\varkappa +1)+\sum _{k=1}^{\infty }{\displaystyle \frac{k^{-2\varkappa -1}}{e^{2\alpha k}-1}}\right\}\hfill \\ \hfill & \hspace{0.25em}={(-\beta )}^{-\varkappa }\left\{{\displaystyle \frac{1}{2}}\zeta (2\varkappa +1)+\sum _{k=1}^{\infty }{\displaystyle \frac{k^{-2\varkappa -1}}{e^{2\beta k}-1}}\right\}\hfill \\ \hfill & -2^{2n}\sum _{j=0}^{\varkappa +1}{(-1)}^j{\displaystyle \frac{B_{2j}}{\left(2j\right)!}}{\displaystyle \frac{B_{2\varkappa +2-2j}}{(2\varkappa +2-2j)!}}{\alpha }^{\varkappa +1-j}{\beta }^j,\hfill \end{array}$$

(1)

where $\zeta \left(s\right)$ resp. $B_j$ is the Riemann zeta-function resp. the jth Bernoulli number, cf. Section 1. We refer to the most extensive account of historical information pertaining to Ramanujan’s formula, cf. [1,2], etc. As one can see in [3] (§4.5), Ramanujan’s formula (Equation (1)) has been established as an equivalent assertion to the functional equation of Hecke type, cf. Section 2.

Theorem 1

(General Ramanujan’s formula). The upper half-plane version (Hecke modular relation) of [4] ((2.4), (2.5)) reads, with $\tau \in \mathcal{H}$,

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

(2)

where ${\sigma }_{-2\varkappa -1}\left(k\right)$ is the sum-of-divisors function and ϰ is any integer ≥0.

On the other hand, for $\varkappa <0$, we have the automorphy of Eisenstein series

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

(3)

This seems to depend on the fact that Hecke theory is for the cusp forms and to form the Mellin transform of a modular form, its constant term is to be subtracted, as on [5] (p. 103).

Proof is given in Section 2 as a consequence of Corollary 1 to Theorem 2.

The partial fraction expansion (PFE) for the cotangent function took place in the theory of zeta-functions in [6] and later in [7]. The latter was incorporated in [8] (§5.4) as an ingredient of equivalent conditions to the functional equation. In [3] (Corollary 4.5.1), Hamburger’s treatment was incorporated which includes the Poisson summation formula, etc. It turns out that as a modular relation, the cotangent function (with its PFE) is rather remote from the functional equation and its effect is not clearly visible. In this paper we are concerned with the role of the cotangent function (with its PFE) as a version of the polylogarithm function of order 0 in the theory of modular relations. We shall give new proofs of Theorem 1 in the case $\varkappa \ge 0$ and show that the case $\varkappa <0$ leads to the Riesz sums.

Notation and Preliminaries

We use the following symbols freely: The Riemann zeta-function is defined by

$$\zeta \left(s\right)=\prod _p{\left(1-p^{-s}\right)}^{-1}=\sum _{n=1}^{\infty }{\displaystyle \frac{1}{n^s}},$$

absolutely convergent for $\sigma =\mathrm{Re}s>1$. $\zeta \left(s\right)$ is continued to a meromorphic function over the whole plane by the functional equation

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

(4)

Lemma 1.

We define Bernoulli numbers by the power series expansion

$${\displaystyle \frac{z}{e^z-1}}=\sum _{n=0}^{\infty }{\displaystyle \frac{B_n}{n!}}{z}^n=1-{\displaystyle \frac{1}{2}}z+{\displaystyle \frac{1}{12}}{z}^2-{\displaystyle \frac{1}{720}}{z}^4+\cdots ,\left|z\right|<2\pi ,$$

(5)

so that

$$B_1=-{\displaystyle \frac{1}{2}},B_2={\displaystyle \frac{1}{6}},B_4=-{\displaystyle \frac{1}{30}},B_6={\displaystyle \frac{1}{42}},B_8=-{\displaystyle \frac{1}{30}},B_{10}={\displaystyle \frac{5}{66}},\cdots$$

(6)

with odd-indexed ones being 0 save for $B_1$.

Lemma 2

(Zeta-values). The following explicit formulas hold:

$$\zeta \left(0\right)=-{\displaystyle \frac{1}{2}},\zeta (-2k)=0(k\ge 1),$$

(7)

$$\zeta (1-2k)=-{\displaystyle \frac{B_{2k}}{2k}}(k\ge 1),$$

(8)

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

(9)

Proof. 

Equations (7) and (8) are unified into

$$\zeta (-k)={(-1)}^k{B}_{k+1},k\ge 0,$$

(10)

from which Equation (9) follows by Equation (4).

The second equality of Equation (9) is also proved in the proof of Lemma 3. □

Remark 1.

The simplest and the most natural proof of Equation (10) is by [9] (Lemma 1); [8] (§3.2) on the integral representation for the partial sums, which in turn depends on the Euler-Maclaurin summation formula.

Lemma 2 is a plausible reasoning that Euler would have taken to find a prototype of the functional equation for the Riemann zeta-function. It was proved in [10] (pp. 89–90) for $(1-2^{1-s})\zeta \left(s\right)$ for two series: one of them is divergent, and the divergent one is summed by a certain summability method. In our case the divergent series are the partial sums mentioned above.

2. Hecke Theory as RHB

We state Hecke theory, or Hecke correspondence, as part of RHB (Riemann–Hecke–Bochner) correspondence in Lemma 2. For more details on RHB correspondence, we refer to [4], [11] (Chapters 3 & 8) and [3] (§2.12). In the theory of RHB correspondence, or general modular relations, there are a few more equivalent conditions to the functional equation—Fourier-Bessel expansion, Ewald expansion, Riesz sums, etc.

Definition 1.

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

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

(11)

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

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

(12)

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

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

and

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

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

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

uniformly in $\sigma ,{\sigma }_1\le \sigma \le {\sigma }_2,\left|t\right|\to \infty$. We refer to the Dirichet series (Equation (11)) as  Hecke L-functions (HLF)  which satisfy (HFE), Equation (12).

Following Bochner [12], define the residual function

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

(13)

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

In relation to the q-expansion (Laurent expansion) [5] (p. 80), we introduce the modular type functions corresponding to Dirichlet series (Equation (11))

$$f\left(\tau \right)=\sum _{n=1}^{\infty }{a}_n{e}^{Ain\tau }\mathrm{and}g\left(\tau \right)=\sum _{n=1}^{\infty }{b}_n{e}^{Ain\tau },\tau \in \mathcal{H},$$

(14)

which are absolutely convergent in the upper half-plane (UHP) $\mathcal{H}$.

They satisfy the (modular) transformation formula

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

(15)

Theorem 2

(Hecke). That the Dirichlet series (Equation (11)) satisfy the functional equation

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

(16)

with the convexity condition is equivalent to the modular transformation in Equation (15) or the Bochner modular relation

$$\sum _{n=1}^{\infty }{a}_n{e}^{-{\lambda }_{n}x}={\left({\displaystyle \frac{A}{x}}\right)}^r\sum _{n=1}^{\infty }{b}_n{e}^{-{\mu }_n{\displaystyle \frac{A^2}{x}}}+\mathrm{P}\left({\displaystyle \frac{x}{A}}\right)$$

(17)

with $\mathrm{Re}x>0$, i.e., the right half-plane (RHP).

This is a version of Hecke’s epoch-making discovery [13,14].

Corollary 1.

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

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

(18)

where the series is absolutely convergent for $\sigma >{\sigma }_{\phi }:=max\{1,1-\mathrm{Re}\alpha \}$. The zeta-function (18) satisfies the functional equation, which, in the case of α being an odd integer, reduces to the Hecke type

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

(19)

and so by Lemma 2, its equivalent—Bochner modular relation (in its symmetric form)

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

(20)

where for $\alpha =2\varkappa +1>0$, we have

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

(21)

while, for $0>\alpha =-(2\varkappa +1)$ odd, ${\mathrm{P}}_{-2\varkappa -1}\left(2\pi x\right)$ is given by

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

(22)

Indeed, Lemma 2 implies the Bochner modular relation (in asymmetric form)

$$\sum _{n=1}^{\infty }{\sigma }_{-\alpha }\left(n\right)e^{-nx}={(-1)}^{{\displaystyle \frac{\alpha -1}{2}}}{\left({\displaystyle \frac{x}{2\pi }}\right)}^{\alpha -1}\sum _{n=1}^{\infty }{\sigma }_{-\alpha }\left(n\right)e^{-{\displaystyle \frac{4{\pi }^{2}n}{x}}}+{\mathrm{P}}_{\alpha }\left(x\right).$$

(23)

This and Equation (20) are clearly equivalent. The transition between the RHP and UHP is done by the substitution

$$x=-i\tau .$$

This proves Theorem 1.

Writing $\alpha =\pi x>0,\beta ={\displaystyle \frac{\pi }{x}}>0$, we may rewrite Equation (1) into the symmetric Bochner modular relation (Equation (20)). On the other hand, the transition between the polylogarithm and the rapidly convergent series is made by Liouville’s formula.

Remark 2.

The functional equation for Equation (18) takes simple form only when α is an odd integer. Even in the case of α being an even integer, the functional equation is involved and we obtain consequences of a more complicated nature, cf. [15].

3. Cotangent Function as the Polylogarithm Function of Order 0

In this section we study the case ${\displaystyle \frac{f^{\prime }}{f}}\left(z\right)=\pi cot\pi z$ and some parts of the contour are shifted to infinity.

We assemble important properties of the cotangent function.

Lemma 3.

We have the global expressions

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

(24)

and the rapidly decreasing expressions

$$\begin{array}{c}\hfill \pi cot\pi z=-\pi i-2\pi i\sum _{n=1}^{\infty }{e}^{2\pi inz}(y>0),\end{array}$$

(25)

$$\begin{array}{c}\hfill \pi cot\pi z=\pi i+2\pi i\sum _{n=1}^{\infty }{e}^{-2\pi inz}(y<0).\end{array}$$

(26)

We also need the Taylor expansion

$$\begin{array}{c}\hfill \pi zcot\pi z=1+\sum _{k=1}^{\infty }{(-1)}^k{B}_{2k}{\displaystyle \frac{{\left(2\pi z\right)}^{2k}}{\left(2k\right)!}}.\end{array}$$

(27)

Proof. 

(27) follows by comparing Equation (5) and the third equality of Equation (24) in the form

$$\begin{array}{c}\hfill \pi zcot\pi z=\pi iz+{\displaystyle \frac{2\pi iz}{e^{2\pi iz}-1}},\left|z\right|<1.\end{array}$$

The first equality of (9) follows by comparing Equation (27) and another form of the Taylor expansion

$$\begin{array}{c}\hfill \pi zcot\pi z=1-2\sum _{k=1}^{\infty }\zeta \left(2k\right)z^{2k},\left|z\right|<1\end{array}$$

(28)

which follows from the PFE Equation (24) for the cotangent function by substituting ${\displaystyle \frac{1}{1-{\left(\frac{z}{n}\right)}^2}}={\sum }_{k=0}^{\infty }{\displaystyle \frac{z^2}{n^2}}$. □

In the following corollaries, we consider the special situation of Theorem 4, which we state as

Lemma 4.

Suppose that $f\left(z\right)$ is meromorphic in $D=\{z=x+iy|\alpha \le x\le \beta ,\left|y\right|\le \gamma \}$, where it has finitely many poles at $z_1,\cdots ,z_p$. $\alpha <\beta$, $\gamma >0$, $\left[\alpha \right]=m\ne \alpha$, $\left[\beta \right]=n\ne \beta$. Let $C=\partial D$ or similar basic contour contained in D and encircling all poles $z_j$. if some of the poles coincide, we compute the residue in each occasion. Let $\mu \ne 0$ be a constant. Then we have

$${\int }_{C}f\left(z\right)\pi cot\pi \mu z\mathrm{d}x=\sum _{k=m}^{n}f\left({\displaystyle \frac{k}{\mu }}\right)+\sum _{k=1}^p\underset{z=z_k}{\mathrm{Res}}f\left(z\right)\pi cot\pi \mu z.$$

(29)

In view of Equation (24), the pick-up function may be changed from $\pi cot\pi \mu z$ into ${\displaystyle \frac{2\pi i}{e^{2\pi i\mu z}-1}}$ with $-\pi i{\sum }_{k=1}^p{\mathrm{Res}}_{z=z_k}f\left(z\right)$ added to the right-hand side.

The following corollary is not directly related to our main scope but we give a proof since it is one of modular relations á la Hamburger mentioned above.

Corollary 2

(Poisson summation formula). If $f\left(z\right)$ is holomorphic on $\{z=x+iy|\left|y\right|\le \gamma \}$ for $\gamma >0$ and

$${\int }_{-\infty }^{\infty }\underset{\left|y\right|\le \gamma }{max}\left|f(x+iy)\right|\mathrm{d}x<\infty .$$

(30)

Further suppose that $f\in C^2\left(\mathbb{R}\right)$, ${\int }_{-\infty }^{\infty }f\left(t\right)\mathrm{d}t$ exists and $f^{″}\in L^1$. Then for $u\in \mathbb{R}$ we have

$$\sum _{n=-\infty }^{\infty }f(n+u)=\sum _{n=-\infty }^{\infty }\widehat{f}\left(n\right)e^{2\pi inu},$$

(31)

where

$$\widehat{f}\left(n\right)={\int }_{-\infty }^{\infty }{e}^{-2\pi int}f\left(t\right)\mathrm{d}t$$

is the Fourier transform of f and the left-hand side is to mean the mean at discontinuities.

Proof. 

This is the case where we consider the rectangle whose vertical sides go to 0 as $\nu \to \infty$. This is proved in detail in [16] (pp. 113–119) with all necessary details. f is assumed not only to be holomorphic but also to satisfy the estimate $f\left(z\right)=O\left({\displaystyle \frac{1}{x^2+1}}\right)$. Hence Equation (30) and the integral $I_n$ being absolutely convergent, is given the meaning of the Fourier transform $\widehat{f}\left(n\right)$.

$$\sum _{n\in \mathbb{Z}}f(n+u)={\int }_{L_1}{\displaystyle \frac{f(z+u)}{e^{2\pi iz}-1}}-{\int }_{L_2}{\displaystyle \frac{f(z+u)}{e^{2\pi iz}-1}},$$

(32)

where $L_1:z=x-i\gamma ,x\in \mathbb{R}$ and $L_2:z=x+i\gamma ,x\in \mathbb{R}$. Applying Equation (25) resp. Equation (26) to $L_2$ resp. $L_1$, we obtain the rapidly convergent expression. □

Our main result is the following theorem.

Theorem 3.

(i) Ramanujan’s formula in Theorem 1 including the eta transformation Equation (35) is a consequence of the intrinsic properties of the polylogarithm function ${\mathrm{Li}}_0\left(e^{\pi i\tau }\right)$, which in turn are connected with the functional equation for the Riemann zeta-function in an implicit way.

(ii) On the other hand, the automorphy of Equation (3) in Theorem 1 with $\alpha =1-2\varkappa <0$ does not follow from the above procedure but amounts to Riesz typical means, making recourse to the case $\alpha =0$.

3.1. Proof of Ramanujan’s Formula in the Case $\varkappa >0$

Proof. 

Proof of Theorem 3 in the case $\varkappa >0$ by a standard method. Since Berndt generalized this by the method of Siegel, we give the familiar procedure of taking limits in the contour integral. Choose in Lemma 4

$$f\left(z\right)=f_{\varkappa }\left(z\right)={\displaystyle \frac{1}{z^{2\varkappa +1}}}\pi cot\pi z/\tau$$

(33)

for $\varkappa \in \mathbb{N}\cup \left\{0\right\}$ and integrate $f\left(z\right)\pi cot\pi z$ along the rectangle $C_{\nu }$ with vertices at $\nu \pm i\nu$, $-\nu \pm i\nu$, with $\nu =N+{\displaystyle \frac{1}{2}}$ where N is a large integer. The horizontal integrals converge to 0 as $\nu \to \infty$ in view of the rapidly convergent factor. The vertical integrals may be shifted, except in the case $\varkappa =0$, in view of the factor ${\displaystyle \frac{1}{z^{2\varkappa +1}}}$ and we are left with the residual functions, which give rise to Equation (2) up to the remaining integral in the case $\varkappa =0$ which will be treated in the next corollary. The right-hand side of Equation (29) becomes $2{\sum }_{k=1}^{\infty }{\displaystyle \frac{1}{k^{2\varkappa +1}}}\left(cot\pi k/\tau +cot\pi k\tau \right)$ plus the residue at $z=0$. It is crucial that Equation (33) is an even function for this resulting sum to be rapidly convergent. Then we use Lemma 3 to choose an appropriate form. □

3.2. Treatment of Theorem 3 in the Case $\varkappa \le 0$

As we have seen above, the choice $f_0$ of integrand and the curve of $C_{\nu }$ causes a difficulty in that one cannot shift the vertical integrals. Siegel’s innovation lies in integrating the function (with the augmented argument) in Equation (37) along the fixed contour C consisting of sides of the rhombus with vertices at $\pm 1,\pm \tau$ and then taking the limit $\nu \to \infty$. In this section we use a generalization of Siegel’s method to prove that the case $\varkappa <0$ as well as $\varkappa =0$ leads to the following Corollary:

Corollary 3

(eta transformation formula). For τ in the upper half-plane and for $\varkappa \le 0$, Siegel’s method leads to

$$\sum _{k=1}^{\infty }\sum _{\ell =1}^{\infty }{\displaystyle \frac{1}{k}}\left(e^{2\pi ik\ell \tau }-e^{-2\pi ik\ell {\tau }^{-1}}\right)={\displaystyle \frac{\pi i}{12}}(\tau +{\tau }^{-1})+{\displaystyle \frac{1}{2}}log{\displaystyle \frac{\tau }{i}},$$

(34)

which is the eta transformation formula

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

(35)

where $\eta \left(\tau \right)$ is the  Dedekind eta-function

$$\eta \left(\tau \right)=e^{{\displaystyle \frac{\pi i\tau }{12}}}\prod _{n=1}^{\infty }(1-e^{2\pi in\tau }),\mathrm{Im}\tau >0.$$

(36)

Proof. 

Proof goes along the similar lines as Siegel’s proof [17] (cf. also [18]). Let $\tau =it\in \mathcal{H}$ be fixed. We choose ($\varkappa \ge 0$)

$$g_{\varkappa }\left(\nu z\right):=z^{2\varkappa -1}\pi cot\left(\pi \nu z\right)\pi cot(\pi \nu z/\tau )$$

(37)

and integrate this along the fixed contour S consisting of sides of the rhombus with vertices at $\pm 1,\pm \tau$ and then take the limit $\nu \to \infty$. Because of the choice of the integrand, the poles are at $z=\pm {\displaystyle \frac{k}{\nu }}$ (this is the picking-up part) and $z=\pm {\displaystyle \frac{k\tau }{\nu }}$$(1\le k\le \nu )$ and the triple pole at $z=0$. Hence, correspondingly to (39),

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2\pi i}}{\int }_S{g}_{\varkappa }\left(\nu z\right)\mathrm{d}z\hfill \\ \hfill & \hspace{0.25em}=2\pi \left(\sum _{k=1}^{h\left(\nu \right)}{\left({\displaystyle \frac{k}{\nu }}\right)}^{2\varkappa -1}{\displaystyle \frac{1}{\nu }}cot{\displaystyle \frac{\pi k}{\tau }}+\sum _{k=1}^{h\left(\nu \right)}{\left({\displaystyle \frac{k\tau }{\nu }}\right)}^{2\varkappa -1}{\displaystyle \frac{\tau }{\nu }}cot\pi \tau k\right)+{\mathrm{P}}_0\hfill \\ \hfill & \hspace{0.25em}=2\pi \left(\sum _{k\le \nu }{\left({\displaystyle \frac{\nu -k}{\nu }}\right)}^{2\varkappa }{\displaystyle \frac{1}{k}}cot{\displaystyle \frac{\pi k}{\tau }}+{(-1)}^{\varkappa }\sum _{k\le {\displaystyle \frac{\nu }{t}}}{\left({\displaystyle \frac{\frac{\nu }{t}-k}{\frac{\nu }{t}}}\right)}^{2\varkappa }{\displaystyle \frac{1}{k}}cot\pi \tau k\right)+{\mathrm{P}}_0+o\left(1\right),\hfill \end{array}$$

(38)

where

$${\mathrm{P}}_0=\underset{z=0}{\mathrm{Res}}{g}_{\varkappa }\left(\nu z\right)=\left\{\begin{array}{cc}-{\displaystyle \frac{{\pi }^2}{3}}(\tau +{\tau }^{-1})\hfill & \varkappa =0\hfill \\ {\displaystyle \frac{\tau }{{\nu }^2}}\pi \hfill & \varkappa =1\hfill \end{array}\right.$$

and ${\mathrm{P}}_0=0$ for $\varkappa >1$ and where $h\left(\nu \right)$ is determined such that $k\le \nu$, $k\le {\displaystyle \frac{\nu }{t}}$, so that $h\left(\nu \right)\to \infty$. Here

$$\begin{array}{c}\hfill \sum _{k\le \nu }{\left({\displaystyle \frac{\nu -k}{\nu }}\right)}^{2\varkappa }{\displaystyle \frac{1}{k}}cot{\displaystyle \frac{\pi k}{\tau }},\sum _{k\le {\displaystyle \frac{\nu }{t}}}{\left({\displaystyle \frac{\frac{\nu }{t}-k}{\frac{\nu }{t}}}\right)}^{2\varkappa }{\displaystyle \frac{1}{k}}cot\pi \tau k\end{array}$$

are Riesz sums, so that their limits as $\nu \to \infty$ are Riesz typical means. Since typical means is a regular method of summation [19], Equation (38) amounts to the $\varkappa =0$ case, the limit of Equation (39).

Thus it suffices to consider

$${\displaystyle \frac{1}{2\pi i}}{\int }_{S}g\left(\nu z\right)\mathrm{d}z=2\pi \left(\sum _{k=1}^{h\left(n\right)}{\displaystyle \frac{1}{k}}cot\pi {\displaystyle \frac{\pi k}{\tau }}+\sum _{k=1}^{h\left(n\right)}{\displaystyle \frac{1}{k}}cot\pi \tau k\right)-{\displaystyle \frac{{\pi }^2}{3}}(\tau +{\tau }^{-1}),$$

(39)

or by Equation (24)

$${\displaystyle \frac{1}{8}}{\int }_{S}g\left(\nu z\right)\mathrm{d}z=\sum _{k=1}^{h\left(n\right)}{\displaystyle \frac{1}{k}}\left({\displaystyle \frac{1}{e^{-2\pi ik\tau }-1}}-{\displaystyle \frac{1}{e^{2\pi ik/\tau }-1}}\right)-{\displaystyle \frac{\pi i}{12}}(\tau +{\tau }^{-1}),$$

(40)

Now we take the limit of the integrand as $\nu \to \infty$, and the limit is ${\displaystyle \frac{{\pi }^2}{z}},-{\displaystyle \frac{{\pi }^2}{z}},{\displaystyle \frac{{\pi }^2}{z}},-{\displaystyle \frac{{\pi }^2}{z}}$ along the sides (and the convergence is bounded). Thus we arrive at integration of $\pm {\displaystyle \frac{{\pi }^2}{z}}$ along S. Then the integral along S may be computed to be $4{\pi }^{2}log{\displaystyle \frac{\tau }{i}}$ as in Siegel [17] or in [18] (pp. 155–157), whence Equation (2). □

3.3. Comparison with Berndt’s Procedure

Generalization of Equation (1) ($\varkappa >0$) is done in [20] (pp. 429–432, Entry 20). We make clear that it is consistent with Theorem 3 in the case $\varkappa >0$. Berndt’s situation is as follows. Let $\alpha ,\beta ,r>0$ with $\alpha \beta =\pi$ and $\alpha /\beta =r$ and let $\phi \left(z\right)$ be an entire function. Then let

$$f_N\left(z\right)={\displaystyle \frac{\phi \left(2\beta \nu z\right)}{z^{2\varkappa +1}\left(e^{2\pi i\nu z/r}-1\right)\left(e^{-2\pi \nu z}-1\right)}}$$

(41)

where $\nu =N+{\displaystyle \frac{1}{2}}$, N is a positive integer and B is the parallelogram with vertices at $\pm i,\pm r$. It is assumed that

$$g_N\left(z\right):=f_N\left(z\right)/{\nu }^{2\varkappa }\to 0,N\to \infty$$

(42)

boundedly on $\mathbb{C}\setminus \{\pm i,\pm r\}.$ The procedure is to integrate $g_N\left(z\right)$ along B and let $N\to \infty$ to deduce a relation among residues, which gives a generalization of Equation (1). This corresponds to our situation in Equation (33) with $f\left(z\right)={\displaystyle \frac{\phi \left(2\beta \nu z\right)}{z^{2\varkappa +1}\left(e^{2\pi i\nu z/r}-1\right)}}$, ${\displaystyle \frac{g^{\prime }}{g}}\left(z\right)\sim {\displaystyle \frac{1}{e^{-2\pi \nu z}-1}}$ and with the contour $C_{\nu }$ (the rectangle with vertices at $\pm \nu r,\pm i\nu$). Thus, this is rather a real half-plane version of Siegel et al.’s method and is consistent with Theorem 3. Since $f_{\nu }$ is assumed to have a pole of order $2\varkappa +3$ at the origin, the statement is to be added that $\phi \left(z\right)$ has no zero at the origin.

Ref. [21] (pp. 253–254, Entry 8) contains a statement that looks related to Corollary 3. In Berndt, the condition is assumed corresponding to Equation (42) with $\varkappa =0$, which, compared with the limit behavior of Equation (37), implies that his theorem has no relevance to Corollary 3. Since it is assumed that there is a triple pole at $z=0$, the additional condition needs to be added that $\phi \left(0\right)\ne 0$.

Table 1. Integrand for Ramanujan’s formula.

Name	Siegel	Berndt
cotangent	$\pi cot\left(\pi \nu z\right)\pi cot(\pi \nu z/\tau )$	$-\pi icoth\left(\pi \nu iz\right)\pi cot(\pi \nu z/\tau )$
Lambert	${\displaystyle \frac{1}{\left(e^{2\pi \nu z}-1\right)\left(e^{-2\pi i\nu z/t}-1\right)}}$	${\displaystyle \frac{1}{\left(e^{-2\pi \nu z}-1\right)\left(e^{2\pi i\nu z/t}-1\right)}}$
polylogarithm	${\displaystyle \frac{1}{\left({\mathrm{Li}}_0\left(e^{2\pi \nu z}\right)+1\right)\left({\mathrm{Li}}_0\left(e^{-2\pi i\nu z/t}\right)+1\right)}}$	${\displaystyle \frac{1}{\left({\mathrm{Li}}_0\left(e^{-2\pi \nu z}\right)+1\right)\left({\mathrm{Li}}_0\left(e^{-2\pi i\nu z/t}\right)+1\right)}}$
formula	Hecke	Bochner

compares [20] and [17].

One of the reasons why the cotangent function has so many built-in properties related to the modular relation is that it arises from the 0th polylogarithm function

$${\mathrm{Li}}_0\left(z\right)={\displaystyle \frac{z}{1-z}}=\sum _{n=1}^{\infty }{z}^n,\left|z\right|<1,$$

(43)

or the geometric series. Its uniformization

$$\begin{array}{c}\hfill {\mathsf{\ell }}_0\left(\tau \right)={\mathrm{Li}}_0\left(e^{2\pi i\tau }\right)={\displaystyle \frac{e^{2\pi i\tau }}{1-e^{2\pi i\tau }}}={\displaystyle \frac{1}{2}}\left(-1+icot\pi \tau \right)\end{array}$$

(44)

is also ubiquitous. We state another related formula

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{e^{2\pi i\tau }-1}}=-{\mathrm{Li}}_0\left(e^{2\pi i\tau }\right)-1.\end{array}$$

(45)

4. A Chapter in Complex Analysis Governed by Argument Principle

In this section we exhibit a plausible self-contained chapter in complex analysis, which is governed by the general argument principle below, up to some auxiliary results. Cf. [22,23], etc.

Theorem 4

(General argument principle). Suppose $f\left(z\right)$ and $g\left(z\right)$ are (one-valued) meromorphic functions in a domain D and $C\subset D$ is a closed curve on whose boundary there is no zero or pole of f and g. Suppose inside C, $f\left(z\right)$ has finitely many poles $z_i$, $1\le i\le k$ with multiplicity $p_i$ and finitely many zeros ${\alpha }_j$, $1\le j\le l$ with multiplicity $n_j$ and that $g\left(z\right)$ has finitely many poles ${\zeta }_k$, $1\le k\le n$ (none of the poles or zeros coincide; if some of them coincide, we compute the residue on each occasion). Then, we have

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

(46)

This has the following implications:

• Argument principle. This is $g\left(z\right)=1$ case. ⟹
• Rouche’s theorem. ⟹
• Open mapping theorem. ⟹
• Maximum modulus principle. ⟹
• Phragmén-Lindelöf principle. ⟹

Corollary 4

(Integral expression for the inverse function). This is $g\left(z\right)=z$, $f\left(z\right)-w$ case. Suppose $f\left(z\right)$ is analytic in a domain D containing $z=0$ and that $f\left(0\right)=0$ and $f^{\prime }\left(0\right)\ne 0$. Prove that the inverse function $g\left(w\right)$ exists in a neighborhood of $w=0$, i.e., the unique solution of the equation $w=f\left(z\right)$ and that we have the explicit formula for $g\left(w\right)$

$$g\left(w\right)={\displaystyle \frac{1}{2\pi i}}{\int }_{C}z{\displaystyle \frac{f^{\prime }\left(z\right)}{f\left(z\right)-w}}\mathrm{d}z,$$

(47)

where C is a certain circle around the origin.

Here, we need the following:

Lemma 5.

Let D be an open set not containing ∞ and let C be a rectifiable Jordan arc in $\mathbb{C}$. Suppose the function $\phi (z,w)$ satisfies the two conditions. (a) $\phi (z,w)$ is continuous on $D\times C$ and (b) for every fixed $w\in C$, $\phi (z,w)$ is analytic in D. Then, $f\left(z\right)={\int }_C\phi (z,w)\mathrm{d}w$ is analytic in D.

Corollary 5

(Lagrange). (i) Suppose $f\left(z\right)$ is meromorphic in a domain D and $C\subset D$ is a closed curve on whose boundary there is no zero or pole of f and that $g\left(z\right)$ is holomorphic in $D\cup C$. Suppose $f\left(z\right)$ has finitely many poles $z_i$, $1\le i\le k$ with multiplicity $p_i$ and that it has finitely many zeros ${\alpha }_j$, $1\le j\le l$ with multiplicity $n_j$. Then, prove that

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

(48)

(ii) Suppose $f\left(z\right)$ is analytic in a domain D and $f^{\prime }\left(z_0\right)\ne 0$ for $z_0\in D$. Then, the inverse function $z=g\left(w\right)$ exists in a neighborhood of $w_0=f\left(z_0\right)$. Prove that the Taylor expansion holds in the following form:

$$z=g\left(w\right)=z_0+\sum _{n=1}^{\infty }{\displaystyle \frac{b_n}{n!}}{(w-w_0)}^n,$$

(49)

where

$$b_n=\underset{z\to z_0}{lim}{\displaystyle \frac{{\mathrm{d}}^{n-1}}{\mathrm{d}{z}^{n-1}}}\left(h{\left(z\right)}^n\right),h\left(z\right)={\displaystyle \frac{z-z_0}{f\left(z\right)-w_0}}.$$

(50)

Thanks to one of the referees, we are able to add many relevant references including [24,25,26]. Refs. [8,14–16,23,35,45,60,62–65,69,70,125] of [25] are suggested references. Hopefully, this will open a new research area.