The Generalized Eta Transformation Formulas as the Hecke Modular Relation

1. Hecke Modular Relation for Generalized Eta-Functions

Rademacher’s “Topics” [1], along with Siegel’s “Advanced analytic number theory”[2], has been the masterpiece classic of the theory of algebraic aspect of analytic number theory and widely read by researchers. [1][Chapter 9] is devoted to the theory of the transformation formula for the Dedekind eta-function $\eta \left(\tau \right)$; hereafter abbreviated as ETF. The main concern is about the ETF under a general Möbius transformation, not restricted to the Spiegelung $S:\tau \to {\tau }^{-1}$. The correspondence between the transformation formula under the Spiegelung and the functional equation for the associated zeta-, L-functions has been known as the Hecke correspondence or more generally as the Riemann-Hecke-Bochner correspondence, RHB correspondence, also referred to as modular relation. This is developed by many authors [3,4,5,6,7,8,9,10,11,12], culminated by [13].

Rademacher [1][Chapter 9], however, incorporates Iseki’s paper [14] for the proof of ETF under a general substitution. [14] depends on the partial fraction expansion (PFE) for the cotangent function and [1] gives an impression that ETF must be proved by PFE. But it is known that PFE is equivalent to the functional equation for the Riemann zeta-function $\zeta \left(s\right)$, [15], which naturally implies that ETF is also a consequence of RHB correspondence. Indeed, Rademacher himself [16] developed the integral transform method to prove ETF prior to Hecke’s discovery of RHB correspondence and his method was used by many subsequent authors [17,18,19,20,21], et al. all of whom used Rademcaher’s method not RHB correspondence. Iseki [22] seems to be the first who revived Rademacher’s method [16] to prove the functional equation, which was extended to the case of Lambert series by Apostol [23]. Both used the gamma transform (56) of the Estermann type zeta-function but RHB correspondence does not seem to be perceived.

Thus the real starter of the proper use of RHB correspondence is [24],which cites [5] and proves the general ETF from the generating zeta-function satisfying the ramified (Hecke) functional equation. [25], a sequel to [24] treats a more general eta-function on a totally real field of degree n by similar argument based on RHB correspondence. On the other hand, [26] adopted RHB correspondence, streamlining [20] and [21].

Our main aim is to elucidate the (Hecke) modular relation structure involved in earlier works by Rademacher, Dieter, Schoenberg et al. and make further developments. In this paper we confine ourselves to the case of Lambert series but as we will see, there appear the Koshlyakov transforms which are used recently, cf. [27].

Notation and symbols. Let

$${\ell }_s\left(x\right)=\sum _{n=1}^{\infty }\frac{e^{2\pi inx}}{n^s},\sigma >1,x\in \mathbb{R}\mathrm{or}\mathrm{Im}x>0,s\in \mathbb{C}$$

be the Lerch zeta-function and

$$\zeta (x,x)=\sum _{n=0}^{\infty }\frac{1}{{(n+x)}^s}0<x\le 1$$

the Hurwitz zeta-function, respectively. For $x=1$ (and $\sigma >1$), they reduce to the Riemann zeta-function

$$\zeta \left(s\right)=\sum _{n=1}^{\infty }\frac{1}{n^s}\sigma =\mathrm{Re}s>1.$$

We make use of the vector space structures in the scone variable x of both these functions for which we refer to [28,29,30]. Let $C\left(s\right)=\left\{a\right(n\left)\right\}$ be the vector space of all periodic arithmetic functions with period $c(\in \mathbb{N})$ and let $D\left(c\right)$ be the corresponding space of Dirichlet series $f\left(s\right)={\sum }_{n=1}^{\infty }\frac{a\left(n\right)}{n^s}$ both of dimension c. It is shown that one basis of $C\left(c\right)$ is the set of characters and the other is their orthogonality relation, which yields the bases of $D\left(c\right)$: $\left\{{\ell }_s\left(\frac{\nu }{c}\right)\right|\nu =1,\cdots ,c\}$ and $\left\{\zeta \left(s,\frac{\nu }{c}\right)\right|\nu =1,\cdots ,c\}$, respectively. One of the base change formulas

$$\begin{array}{cc}\hfill & {\ell }_z\left(\frac{\nu }{c}\right)=c^{-z}\sum _{\lambda =1}^c{e}^{2\pi i\frac{\nu }{c}\lambda }\zeta \left(z,\frac{\lambda }{c}\right).\hfill \end{array}$$

(1)

will play an important role.

${\ell }_1\left(x\right)$ is not defined at integer points x and needs separate consideration. E.g. its odd part

$$\frac{1}{2}\left({\ell }_1\left(x\right)-{\ell }_1(1-s)\right)=-\pi i{\overline{B}}_1\left(x\right)$$

(2)

is discontinuous at integer points x but has the value 0. The same applies to ${\ell }_0\left(x\right)$.

Another important vector space is the space ${\mathcal{K}}_s$ of Kubert functions which are periodic functions with period 1 satisfying the Kubert relation $\left({*}_s\right)$:

$$\sum _{r=0}^{m-1}f\left(\frac{x+r}{m}\right)=m^{1-s}f\left(x\right).$$

(*_s)

cf., Milnor [31]. ${\mathcal{K}}_s$ is of dimension 2 and is spanned by ${\ell }_s\left(x\right)$ and ${\ell }_s(1-x)$ for $s\ne$ negative integers while by $\zeta (s,x)$ and $\zeta (s,1-x)$ for $s\ne$ non-negative integers. The Kubert relations

$$\begin{array}{cc}\hfill & \sum _{\mu =1}^c{\ell }_s\left(\frac{x+\mu }{c}\right)=c^{1-s}{\ell }_s\left(x\right),0<x<1\hfill \\ \hfill & \sum _{\mu =1}^c\zeta \left(s,\frac{x+\mu }{c}\right)=c^{1-s}\zeta (s,x),0<x\le 1\hfill \end{array}$$

(3)

hold for $s\in \mathbb{C}$ except for singularities.

Since every element of ${\mathcal{K}}_s$ is a linear combination of these two zeta-functions, we write

$$f(s,x)\leftrightarrow \zeta (s,x),g(s,x)\leftrightarrow {\ell }_s\left(x\right)$$

to mean that $f(s,x)$ is of Hurwitz zeta-type resp. $g(s,x)$ of Lerch zeta-type satisfying the same conditions as $\zeta (s,x)$ resp. ${\ell }_s\left(x\right)$ does. This in particular applies to their even and odd parts.

Define

$${\mathcal{E}}_c^{a,b}(f,g,w,z)=\sum _{\lambda =1}^{c}f\left(w,1-\left\{\frac{a\lambda }{c}\right\}\right)g\left(z,1-\left\{\frac{b\lambda }{c}\right\}\right).$$

(4)

(4) is Estermann’s type of Dedekind sum whose concrete case will appear in the second proof of Theorem 1. We substitute the functional equation

$$f\left(1-w,x\right)=\frac{\Gamma \left(w\right)}{{\left(2\pi \right)}^w}\left(e^{-\frac{\pi i}{2}w}g(w,x)+e^{\frac{\pi i}{2}w}g(w,1-x)\right)$$

or

$$g\left(1-z,x\right)=\frac{\Gamma \left(z\right)}{{\left(2\pi \right)}^z}\left(e^{-\frac{\pi i}{2}z}f(z,1-x)+e^{\frac{\pi i}{2}z}f(z,x)\right).$$

as the case may be to deduce

$$\begin{array}{cc}\hfill & f(1-w,x)g(1-z,y)=\frac{\Gamma \left(w\right)\Gamma \left(z\right)}{{\left(2\pi \right)}^{w+z}}(e^{-\frac{\pi i}{2}(w+z)}f(w,1-x)g(z,y)\hfill \\ \hfill & +e^{\frac{\pi i}{2}(w+z)}f(w,x)g(z,1-y)+e^{\frac{\pi i}{2}(w-z)}f(w,x)g(z,y)\hfill \\ \hfill & +e^{-\frac{\pi i}{2}(w-z)}f(w,1-x)g(z,1-y))\hfill \end{array}$$

(5)

This will appear in §5.

It is Mikolás [32] who first introduced the transcendental generalization of the Dedekind sums in which instead of (4), the $f,f$-type zeta functions are considered as with almost all preceding papers. In the second proof of Theorem 1, we will reveal that the Estermann type zeta-functions makes things simpler.

2. The Rademacher-Apostol Case

In this section we illustrate the elucidation of Rademacher’s integral transform method by showing the functional equation for the zeta-function and the general ETF as developed in Rademacher [16] (for eta function) and also by Apostol [17] (for Lambert series). The residual function in Theorem 1 is the corrected form of that of [17] in the form nearest to Apostol’s. This corrected form was first proved by Mikolás [33][p.106] and shortly thereafter by Iseki [14], both of whom treated the case $p\ge 1$. Then as stated above, [22] proved the Hecke functional equation in the case $p=1$ and Apostol [23] used the same method to treat the case $p>1$, without mentioning RHB correspondence.

Toward the end we shall briefly explain the case of Krätzel [34].

Let $c\in \mathbb{N}$, $p\ge 1$ be an odd integer and let h be an integer such that $(h,c)=1$. Define the Rademacher-Apostol zeta-function

$$Z_p(s,h)=\sum _{\mu ,\nu =1}^c{e}^{\frac{2\pi ih\mu \nu }{c}}\zeta \left(s,\frac{\mu }{c}\right)\zeta \left(s+p,\frac{\nu }{c}\right).$$

(6)

Let

$$g_p\left(x\right)=g_p\left(e^{2\pi i\frac{iz+h}{c}}\right)=\frac{1}{2\pi i}{\int }_{\left(\gamma \right)}\Gamma \left(s\right)Z_p(s,h)c^{-1}{\left(2\pi cz\right)}^{-s}\mathrm{d}s,$$

(7)

be the Hecke gamma transform of $Z_p(s,h)$ as in [16][(1.14)], where $\gamma >1$.

Theorem 1.

The zeta-function $Z_p(s,h)$ satisfies the Hecke functional equation

$${\left(2\pi c\right)}^{-s-\frac{p-1}{2}}\Gamma \left(s\right)Z_p(s,h)={\left(2\pi c\right)}^{s+\frac{p-1}{2}}{(-1)}^{\frac{p-1}{2}}\Gamma (-s)Z_p(1-p-s,H),$$

(8)

where H is an integer such that

$$hH\equiv -1modc.$$

(9)

. The Lambert series (7) satisfies the transformation formula

$$g_p\left(e^{2\pi i\frac{iz+h}{c}}\right)=g_p\left(e^{2\pi i\frac{i{z}^{-1}+H}{c}}\right)+{\mathrm{P}}_p\left(z\right),$$

(10)

where

$$\begin{array}{cc}\hfill {\mathrm{P}}_p\left(z\right)& ={\mathrm{Res}}_{s=-p,\cdots ,0,1}\Gamma \left(s\right)Z_p(s,h)c^{-p}{\left(2\pi cz\right)}^{-s}\hfill \\ \hfill & =\frac{-1}{2(p+1)!}{\left(\frac{2\pi z}{c}\right)}^p{B}_{p+1}+\frac{{(-1)}^{\frac{p-1}{2}}}{2(p+1)!}{\left(\frac{2\pi }{c}\right)}^p{z}^{-1}{B}_{p+1}\hfill \\ \hfill & +\frac{-i{\left(2\pi i\right)}^p}{2\left(p\right)!}{s}_{p,1}(c,h)+\frac{1}{2}{\delta }_{p,1}loga+\frac{1}{2}\left(1-{(-1)}^{\frac{p-1}{2}}\right)\zeta \left(p\right)\hfill \\ \hfill & +\sum _{r=2}^p\frac{{(-1)}^r}{r!}{\left(2\pi z\right)}^{r-1}\frac{-{\left(2\pi i\right)}^{p+1-r}}{2(p+1-r)!}{s}_{p,r}(c,h),\hfill \end{array}$$

(11)

and where ${\delta }_{p,1}$ is the Kronecker symbol.

Proof. 

We combine the Hurwitz formula (12) and the base change formula (13) with $f={\chi }_{\mu }$ to deduce (14): The Hurwitz formula (i.e., the functional equation for the Hurwitz zeta-function): for $\sigma >1,0<x\le 1$,

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

(12)

The base change—linear combination expression—formula reads

$$\begin{array}{cc}\hfill & \frac{1}{c^s}\sum _{a=1}^{c}a\left(n\right)\zeta \left(s,\frac{n}{c}\right)=D(s,a)=\frac{1}{\sqrt{c}}\sum _{n=1}^c\widehat{a}\left(n\right){\ell }_s\left(\frac{n}{c}\right)\hfill \\ \hfill & =\frac{1}{\sqrt{c}}\sum _{n=1}^{c-1}\widehat{a}\left(n\right){\ell }_s\left(\frac{n}{c}\right)+\frac{\widehat{a}\left(c\right)}{\sqrt{c}}\zeta \left(s\right),\hfill \end{array}$$

(13)

where $\widehat{a}\left(n\right)$ is the DFT (discrete Fourier transform) of $a\left(n\right)$. Choosing $a\left(n\right)={\chi }_{\mu }\left(n\right)$, $\chi \mu$ being the characteristic function of $\mu$, we see that its DFT is the character, which implies (1).

Combining (12) and (1), we deduce

$$\begin{array}{cc}\hfill & \zeta \left(s,\frac{\mu }{c}\right)=\Gamma (1-s)\frac{2}{{\left(2\pi c\right)}^{1-s}}\hfill \\ \hfill & \times \left(sin\frac{\pi }{2}s\sum _{\lambda =1}^{c}cos\frac{2\pi \lambda \mu }{c}\zeta \left(1-s,\frac{\lambda }{c}\right)+cos\frac{\pi }{2}s\sum _{\lambda =1}^{c}sin\frac{2\pi \lambda \mu }{c}\zeta \left(1-s,\frac{\lambda }{c}\right)\right).\hfill \end{array}$$

(14)

Substituting (14) in (6) and using

$$\begin{array}{cc}\hfill & \sum _{\mu =1}^c{e}^{\frac{2\pi ih\mu \nu }{c}}cos\frac{2\pi \lambda \mu }{c}=\sum _{\mu =1}^{c}cos\frac{2\pi h\mu \nu }{c}cos\frac{2\pi \lambda \mu }{c}\hfill \\ \hfill & \sum _{\mu =1}^c{e}^{\frac{2\pi ih\mu \nu }{c}}sin\frac{2\pi \lambda \mu }{c}=\sum _{\mu =1}^{c}sin\frac{2\pi h\mu \nu }{c}sin\frac{2\pi \lambda \mu }{c},\hfill \end{array}$$

(15)

we conclude that

$$\begin{array}{cc}\hfill Z_p(s,h)& =c^{-1}{\left(2\pi c\right)}^s(\sum _{\lambda ,\mu ,\nu =1}^{c}cos\frac{2\pi h\mu \nu }{c}cos\frac{2\pi \lambda \mu }{c}\frac{1}{cos\frac{\pi }{2}s}\zeta \left(1-s,\frac{\lambda }{c}\right)\zeta \left(p+s,\frac{\nu }{c}\right)\hfill \\ \hfill & +\sum _{\lambda ,\mu ,\nu =1}^{c}sin\frac{2\pi h\mu \nu }{c}sin\frac{2\pi \lambda \mu }{c}\frac{1}{sin\frac{\pi }{2}s}\zeta \left(1-s,\frac{\lambda }{c}\right)\zeta \left(p+s,\frac{\nu }{c}\right)).\hfill \end{array}$$

(16)

Changing s by $1-p-s$ and $\mu$ by $H\mu$, where H is as in (9), then the second factor remains unchanged up to the additional factor ${(-1)}^{\frac{p-1}{2}}$. Hence

$$Z_p(1-p-s,H)={\left(2\pi c\right)}^{1-p-2s}{(-1)}^{\frac{p-1}{2}}{Z}_p(s,h),$$

which is (8).

Substituting (16) in (7), we derive that

$$\begin{array}{cc}\hfill & g_p\left(x\right)=\frac{1}{2c^{p+1}}\hfill \\ \hfill & \times (\sum _{\lambda ,\mu ,\nu =1}^{c}cos\frac{2\pi h\mu \nu }{c}cos\frac{2\pi \lambda \mu }{c}\frac{1}{2\pi i}{\int }_{\left(\gamma \right)}\frac{1}{cos\frac{\pi }{2}s}\zeta \left(1-s,\frac{\lambda }{c}\right)\zeta \left(s+p,\frac{\nu }{c}\right)z^{-s}\mathrm{d}s\hfill \\ \hfill & +\sum _{\lambda ,\mu ,\nu =1}^{c}sin\frac{2\pi h\mu \nu }{c}sin\frac{2\pi \lambda \mu }{c}\frac{1}{2\pi i}{\int }_{\left(\gamma \right)}\frac{1}{sin\frac{\pi }{2}s}\zeta \left(1-s,\frac{\lambda }{c}\right)\zeta \left(s+p,\frac{\nu }{c}\right)z^{-s}\mathrm{d}s),\hfill \end{array}$$

(17)

which is ([16](1.27)).

Shifting the integration path to $\sigma =1-p-\gamma$ and applying (8), we conclude [16][(1.29)], which is (10).

Incorporating the residual function found in [17] with correction calculated in [27], we arrive at the general transformation formula, entailing ETF ([16](1.45)), completing the proof.

Second proof.

We may give a more lucid proof of (8) using the Estermann type Dedekind sum

$$\begin{array}{cc}\hfill & {\mathcal{E}}_c^{a,b}(w,z)=\sum _{\lambda modc}\zeta \left(w,1-\left\{\frac{a\lambda }{c}\right\}\right){\ell }_z\left(1-\left\{\frac{b\lambda }{c}\right\}\right)\hfill \\ \hfill & =\sum _{\lambda =1}^{c-1}\zeta \left(w,\left\{\frac{a\lambda }{c}\right\}\right){\ell }_z\left(\frac{b\lambda }{c}\right)+\zeta \left(w\right)\zeta \left(z\right).\hfill \end{array}$$

(18)

Estermann [35][(19)] established the functional equation

$${\mathcal{E}}_c^{a,1}(s,s)=-2{\left(2\pi \right)}^{2s-2}{\Gamma }^2(1-s)\left(cos\left(\pi s\right){\mathcal{E}}_c^{1,-a}(1-s,1-s)-{\mathcal{E}}_c^{1,a}(1-s,1-s)\right),$$

(19)

which is a special case of the more general functional equation

$$\begin{array}{cc}\hfill & {\mathcal{E}}_c^{a,b}(1-w,1-z)=\frac{2\Gamma \left(w\right)\Gamma \left(z\right)}{{\left(2\pi \right)}^{w+z}}\hfill \\ \hfill & \times \left(cos\frac{\pi }{2}(w+s){\mathcal{E}}_c^{b,-a}(z,w)+cos\frac{\pi }{2}(w-s){\mathcal{E}}_c^{b,a}(z,w)\right).\hfill \end{array}$$

(20)

We consider the sum slightly more general than (6):

$$I_p(w,z,h):=\sum _{\mu ,\nu =1}^c{e}^{\frac{2\pi ih\mu \nu }{c}}\zeta \left(w,\frac{\mu }{c}\right)\zeta \left(z,\frac{\nu }{c}\right)=\sum _{\mu =1}^c\zeta \left(w,\frac{\mu }{c}\right)\sum _{\nu =1}^c{e}^{\frac{2\pi ih\mu \nu }{c}}\zeta \left(z,\frac{\nu }{c}\right).$$

(21)

The inner sum on the right of (21) is $c^z{\ell }_z\left(\frac{h\mu }{c}\right)$ in view of the base change formula (1) becomes

$$I_p(w,z,h)=c^z\sum _{\mu =1}^c\zeta \left(w,\frac{\mu }{c}\right){\ell }_z\left(\frac{h\mu }{c}\right)=c^z{\mathcal{E}}_c^{1,h}(w,z),$$

(22)

which becomes

$$Z_p(s,h)=I_p(s,s+p,h)=c^{s+p}{\mathcal{E}}_c^{1,h}(s,s+p),$$

(23)

on specifying $w=s$, $z-p+s$. Hence, substituting (20) in (22), we deduce that

$$\begin{array}{cc}\hfill & I_p(w,z,h)=c^z\frac{2\Gamma (1-w)\Gamma (1-z)}{{\left(2\pi \right)}^{2-w-z}}\hfill \\ \hfill & \times \left(-cos\frac{\pi }{2}(w+s){\mathcal{E}}_c^{-h,1}(1-z,1-w)+cos\frac{\pi }{2}(w-s){\mathcal{E}}_c^{h,1}(1-z,1-w)\right).\hfill \end{array}$$

(24)

Specifying $w=s$, $z-p+s$, (24) reads

$$\begin{array}{cc}\hfill & Z_p(s,h)=I_p(s,s+p,h)=c^{s+p}\frac{2\Gamma (1-s)\Gamma (1-p-s)}{{\left(2\pi \right)}^{2-2s-p}}\hfill \\ \hfill & \times \left(-cos\frac{\pi }{2}(2s+p){\mathcal{E}}_c^{-h,1}(1-p-s,1-s)+cos\frac{\pi }{2}p{\mathcal{E}}_c^{h,1}(1-p-s,1-s)\right).\hfill \end{array}$$

(25)

Taking oddness of p into accout, this reduces to

$$\begin{array}{c}\hfill Z_p(s,h)=c^{s+p}\frac{2\Gamma (1-s)\Gamma (1-p-s)}{{\left(2\pi \right)}^{2-p-2s}}{(-1)}^{\frac{p-1}{s}}sin\pi s{\mathcal{E}}_c^{-h,1}(1-p-s,1-s),\end{array}$$

whence

$$\begin{array}{c}\hfill \Gamma \left(s\right)Z_p(s,h)=c^{s+p}\frac{\Gamma (1-p-s)}{{\left(2\pi \right)}^{1-p-2s}}{(-1)}^{\frac{p-1}{s}}{\mathcal{E}}_c^{-h,1}(1-p-s,1-s).\end{array}$$

(26)

Now let H be as in (9). Then

$$\begin{array}{c}\hfill {\mathcal{E}}_c^{-h,1}(1-p-s,1-s)={\mathcal{E}}_c^{1,H}(1-p-s,1-s)=c^{1-s}{Z}_p(1-p-s,H)\end{array}$$

by (23). Substituting this in (26) proves (8).

Third proof. We may restore the argument of [16] (and [17]) to prove (10) and the proof entails the proof of (8)., cf. [27]. □

3. The Krätzel Case

[34] deals with a generalization (38) of the eta-function which depends on the Hecke gamma transform of the zeta-function

$$Z_{a.b}\left(s\right):=\frac{1}{\Gamma (s+1)sin\frac{\pi }{2ab}s}\zeta \left(\frac{1}{a}s\right)\zeta \left(-\frac{1}{b}s\right),$$

(27)

where $a,b$ are natural numbers, $(a,b)=1$. $Z_{a.b}\left(s\right)$ satisfies the Hecke functional equation

$$\Gamma \left(s\right)Z_{a.b}\left(s\right)=\Gamma (-s)Z_{b,a}(-s).$$

(28)

Krätzel’s method is essentially that of Rademacher although he does not refer to [16] and we give a brief account on this point.

Theorem 2.

The Krätzel-Rademacher method yields the modular relation (28) as well as the transformation formula

$${\eta }_{a,b}\left(x\right)=x^{-\frac{ab}{2}}{\eta }_{b,a}\left(\frac{1}{x}\right).$$

(29)

Proof. 

For the moment, we work with ($\mathrm{Re}x>0$ and $|argz|<\frac{\pi }{2ab}$)

$${\tilde{\eta }}_{a,b}\left(x\right):=\prod _{m=1}^{\infty }\prod _{\nu =1}^{a-1}\left(1-e^{2\pi i{\epsilon }_{2\mu +1}\left(4a\right)n^{\frac{b}{a}}{x}^b}\right),$$

(30)

where ${\epsilon }_{2\mu +1}\left(4a\right)=e^{2\pi i\frac{2\nu +1}{4a}}$. Then for $\varkappa >\frac{a}{b}$, we have by the Hecke gamma transform

$$log{\tilde{\eta }}_{a,b}\left(x\right)=-\frac{1}{2\pi i}{\int }_{\left(\varkappa \right)}\Gamma \left(s\right)\zeta (s+1)\zeta \left(\frac{b}{a}s\right)\sum _{\nu =1}^{a-1}{\left(2\pi i{e}^{i\frac{2\nu +1}{4a}}\right)}^{-s}{\left(2\pi x^b\right)}^{-s}\mathrm{d}s.$$

(31)

Now the sum becomes

$$\sum _{\nu =1}^{a-1}{\left(i{e}^{2\pi i\frac{2\nu +1}{4a}}\right)}^{-s}=\frac{sin\frac{\pi }{2}s}{sin\frac{\pi }{2a}s}.$$

Hence (31) becomes

$$log{\tilde{\eta }}_{a,b}\left(x\right)=-\frac{1}{2\pi i}{\int }_{\left(\varkappa \right)}\Gamma \left(s\right)\frac{sin\frac{\pi }{2}s}{sin\frac{\pi }{2a}s}\zeta (s+1)\zeta \left(\frac{b}{a}s\right){\left(2\pi x^b\right)}^{-s}\mathrm{d}s.$$

(32)

Now we apply the functional equation only to one factor $\zeta (s+1)$:

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

(33)

Substituting (33) in (32), we obtain

$$log{\tilde{\eta }}_{a,b}\left(x\right)=\frac{1}{2\pi i}{\int }_{\left(\varkappa \right)}\frac{\Gamma \left(s\right)}{\Gamma (s+1)sin\frac{\pi }{2a}s}\pi \zeta (-s)\zeta \left(\frac{b}{a}s\right){\left(x^b\right)}^{-s}\mathrm{d}s.$$

(34)

Note that the factor $\frac{\Gamma \left(s\right)}{\Gamma (s+1)}\mathrm{d}s$ being $\frac{1}{s}\mathrm{d}s$ remains invariant under the change of variable $s\to as$, so that (34) becomes as in Krätzel,

$$log{\tilde{\eta }}_{a,b}\left(x\right)=\frac{1}{2\pi i}{\int }_{\left({\varkappa }_1\right)}\frac{\Gamma \left(s\right)}{\Gamma (s+1)sin\frac{\pi }{2}s}\pi \zeta (-as)\zeta \left(bs\right){\left(x^{ab}\right)}^{-s}\mathrm{d}s,$$

(35)

where ${\varkappa }_1>\frac{1}{b}$. These two are the main ingredients of Krätzel and corresponds to Rademacher’s (17).

Changing the variable $s\to abs$, (35) becomes

$$log{\tilde{\eta }}_{a,b}\left(x\right)=\frac{1}{2\pi i}{\int }_{\left({\varkappa }_2\right)}\Gamma \left(s\right)Z_{a,b}\left(s\right)x^{-s}\mathrm{d}s,$$

(36)

i.e., the Hecke gamma transform of $Z_{a,b}\left(s\right)$, where ${\varkappa }_2>a$. As usual, shifting the integration path to $\sigma =-{\varkappa }_2<-\frac{1}{a}$, we encounter poles and we are to find residues. The resulting integral is the same as (36) with x changed by $\frac{1}{x}$. Krätzel writes [34][p. 116] “Then under the substitution $s\to -s$, the functional equation (28) follows on symmetry grounds” meaning that he proves (28) at this stage.

Krätzel treats (35) and shifts the line to $-{\varkappa }_2<-\frac{1}{a}$ finding the sum of residues

$$-{\gamma }_{a,b}\left(x\right)+{\gamma }_{b,a}\left(\frac{1}{x}\right)+\frac{1}{2}(b-a)log2\pi -\frac{ab}{2}x,$$

(37)

where

$${\gamma }_{a,b}\left(x\right)=\frac{\pi }{sin\frac{\pi }{2a}}\zeta \left(-\frac{b}{a}\right)x^b.$$

Hence defining

$${\eta }_{a,b}\left(x\right)={\left(2\pi \right)}^{\frac{1-b}{2}}{e}^{{\gamma }_{a,b}\left(x\right)}{\tilde{\eta }}_{a,b}\left(x\right),$$

(38)

we conclude (29). □

4. Unification of Rademacher and Dieter Cases

In this section we prove the modular relation structure of the zeta-functions and the general ETFs contained in [16,17,18]. We work in the framework of Dieter with slight modifications. Let $p,d,f,\alpha ,\beta$ be integers satisfying the conditions $p\ge 1$ being odd, $(h,c)=1$, $f\ge 1$, $0<\alpha \le f$. f works as a fixed aixiliary modulus and $d=-h$ in §2. In Dieter’s case, $\alpha ,\beta \not\equiv 0$ mod f is also assumed. Then the Dieter zeta-function is defined by

$$f_{\alpha ,\beta }(s,x)=f_{p,\alpha ,\beta }\left(s,e^{2\pi i\frac{iz+h}{c}}\right)=\sum _{\mu =0}^{c-1}\sum _{\nu =1}^{fc}{e}^{2\pi i\frac{h\mu \nu +\gamma \nu }{c}}\zeta \left(s,\frac{\mu }{c}+\frac{\alpha }{cf}\right)\zeta \left(s+p,\frac{\nu }{cf}\right),$$

(39)

where

$$\gamma (-\alpha ,-\beta )=-\gamma (\alpha ,\beta ),\gamma =\gamma (\alpha ,\beta )=\frac{-h\alpha -c\beta }{f}.$$

(40)

We assume $\gamma (-\alpha ,-\beta )=\gamma (\alpha ,\beta )$ for $\alpha ,\beta \equiv 0$ mod f, which we abbrviate $\gamma (0,0)$. We also assume that $\mu$ varies $1,\cdots ,c$ in the case of $\gamma (0,0)$. Then (39) with $p=1$ amounts to (6). In almost all susequent researches after Rademacher, it is necessary to consider the even part [18][(2,11)], which is

$$g_{\alpha ,\beta }(s,x):=f_{\alpha ,\beta }(s,x)+f_{-\alpha ,-\beta }(s,x).$$

(44)

One speculated reason for this is stated in [27].

Let

$$G_p\left(x\right)=G_p\left(e^{2\pi i\frac{iz+h}{c}}\right)=\frac{1}{2\pi i}{\int }_{\left(\gamma \right)}\Gamma \left(s\right)g_{\alpha ,\beta }(s,x){\left(cf\right)}^{-1}{\left(2\pi cz\right)}^{-s}\mathrm{d}s,$$

(42)

be the Hecke gamma transform, where $\gamma >1$.

Theorem 3.

Rademacher’s transform yields the transformation formula

$$G_{p,\alpha ,\beta }\left(e^{2\pi i\frac{iz+h}{c}}\right)=G_{p,\alpha ,\beta }\left(e^{2\pi i\frac{i{z}^{-1}+H}{c}}\right)+\mathrm{P}\left(z\right),$$

(43)

where

$$\mathrm{P}\left(z\right)=\sum _{s=p,\cdots ,0,1}\mathrm{Res}\Gamma \left(s\right)g_{\alpha ,\beta }(s,x){\left(cf\right)}^{-1}{\left(2\pi cz\right)}^{-s}.$$

(44)

as well as the Hecke functional equation for the even part $g_{\alpha ,\beta }(s,x)$ of the Dieter zeta-function

$${\left(2\pi cf\right)}^{-s-\frac{p-1}{2}}\Gamma \left(s\right)g_{p,\alpha ,\beta }(s,x)={\left(2\pi cf\right)}^{s+\frac{p-1}{2}}{(-1)}^{\frac{p-1}{2}}\Gamma (1-p-s)g_{p,{\alpha }^{\prime },{\beta }^{\prime }}(1-p-s,x),$$

(45)

where H is an integer as in (10) and

$$\left(\begin{array}{c}\alpha \\ \beta \end{array}\right)=\left(\begin{array}{cc}H& c\\ b& -h\end{array}\right)\left(\begin{array}{c}{\alpha }^{\prime }\\ {\beta }^{\prime }\end{array}\right).$$

(46)

The theoem also covers Theorem 1.

Proof. 

We give a proof verbatim to that of Theorem 1. We employ (14) as

$$\begin{array}{cc}\hfill & \zeta \left(s,\frac{\mu }{c}+\frac{\alpha }{cf}\right)=\Gamma (1-s)\frac{2}{{\left(2\pi cf\right)}^{1-s}}(sin\frac{\pi }{2}s\sum _{\lambda =1}^{c}cos2\pi \lambda \left(\frac{\mu }{c}+\frac{\alpha }{cf}\right)\zeta \left(1-s,\frac{\lambda }{cf}\right)\hfill \\ \hfill & +cos\frac{\pi }{2}s\sum _{\lambda =1}^{c}sin2\pi \lambda \left(\frac{\mu }{c}+\frac{\alpha }{cf}\right)\zeta \left(1-s,\frac{\lambda }{cf}\right)).\hfill \end{array}$$

(47)

Substituting (14) in (8), we find that

$$\begin{array}{cc}\hfill & c{\left(2\pi cf\right)}^{-s}\Gamma \left(s\right)f_{\alpha ,\beta }(s,x)\hfill \\ \hfill & =\sum _{\lambda ,\nu =1}^{fc}(\sum _{\mu =0}^{c-1}{e}^{2\pi i\frac{h\mu \nu +\gamma \nu }{c}}cos2\pi \lambda \left(\frac{\mu }{c}+\frac{\alpha }{cf}\right)\frac{1}{cos\frac{\pi }{2}s}\zeta \left(1-s,\frac{\lambda }{cf}\right)\zeta \left(p+s,\frac{\nu }{cf}\right)\hfill \\ \hfill & +\sum _{\mu =0}^{c-1}{e}^{2\pi i\frac{h\mu \nu +\gamma \nu }{c}}sin2\pi \lambda \left(\frac{\mu }{c}+\frac{\alpha }{cf}\right)\frac{1}{sin\frac{\pi }{2}s}\zeta \left(1-s,\frac{\lambda }{cf}\right)\zeta \left(p+s,\frac{\nu }{cf}\right)).\hfill \end{array}$$

(48)

To proceed further with the non-degenerated (48) we need a counterpart of (15) and for this we need to consider the even part [18][(2,11)], which is (41).

Then we are to incorporate

$$\begin{array}{cc}\hfill & \sum _{\mu =0}^{c-1}{e}^{2\pi i\frac{h\mu \nu +\gamma \nu }{c}}cos2\pi \lambda \left(\frac{\mu }{c}+\frac{\alpha }{cf}\right)+\sum _{\mu =0}^{c-1}{e}^{2\pi i\frac{h\mu \nu -\gamma \nu }{c}}cos2\pi \lambda \left(\frac{\mu }{c}-\frac{\alpha }{cf}\right)\hfill \\ \hfill & =\sum _{\mu =0}^{c-1}{e}^{2\pi i\frac{h\mu \nu +\gamma \nu }{c}}cos2\pi \lambda \left(\frac{\mu }{c}+\frac{\alpha }{cf}\right)+\sum _{\mu =0}^{c-1}{e}^{2\pi i\frac{-h\mu \nu -\gamma \nu }{c}}cos2\pi \lambda \left(\frac{\mu }{c}+\frac{\alpha }{cf}\right)\hfill \\ \hfill & =2\left(\sum _{\mu =0}^{c-1}\mathrm{Re}\left(e^{2\pi i\frac{h\mu \nu +\gamma \nu }{c}}\right)cos2\pi \lambda \left(\frac{\mu }{c}+\frac{\alpha }{cf}\right)\right)\hfill \\ \hfill & =2\sum _{\mu =0}^{c-1}cos2\pi \frac{h\mu \nu +\gamma \nu }{c}cos2\pi \left(\lambda \frac{\mu }{c}+\frac{\alpha }{cf}\right)\hfill \end{array}$$

(49)

and

$$\begin{array}{cc}\hfill & \sum _{\mu =0}^{c-1}{e}^{2\pi i\frac{h\mu \nu +\gamma \nu }{c}}sin2\pi \lambda \left(\frac{\mu }{c}+\frac{\alpha }{cf}\right)+\sum _{\mu =0}^{c-1}{e}^{2\pi i\frac{h\mu \nu -\gamma \nu }{c}}sin2\pi \lambda \left(\frac{\mu }{c}-\frac{\alpha }{cf}\right)\hfill \\ \hfill & =-2i\sum _{\mu =0}^{c-1}sin2\pi \frac{h\mu \nu +\gamma \nu }{c}sin2\pi \left(\lambda \frac{\mu }{c}+\frac{\alpha }{cf}\right).\hfill \end{array}$$

(50)

Substituting in (48), we obatin

$$\begin{array}{cc}\hfill & c{\left(2\pi cf\right)}^{-s}\Gamma \left(s\right)g_{\alpha ,\beta }(s,x)\hfill \\ \hfill & =2\sum _{\lambda ,\nu =1}^{fc}\sum _{\mu =0}^{c-1}cos2\pi \frac{h\mu \nu +\gamma \nu }{c}cos2\pi \left(\lambda \frac{\mu }{c}+\frac{\alpha }{cf}\right)\frac{1}{cos\frac{\pi }{2}s}\zeta \left(1-s,\frac{\lambda }{cf}\right)\zeta \left(p+s,\frac{\nu }{cf}\right)\hfill \\ \hfill & -2i\sum _{\lambda ,\nu =1}^{fc}\sum _{\mu =0}^{c-1}sin2\pi \frac{h\mu \nu +\gamma \nu }{c}sin2\pi \left(\lambda \frac{\mu }{c}+\frac{\alpha }{cf}\right)\frac{1}{sin\frac{\pi }{2}s}\zeta \left(1-s,\frac{\lambda }{cf}\right)\zeta \left(p+s,\frac{\nu }{cf}\right).\hfill \end{array}$$

(51)

Changing s by $1-p-s$ and $\mu$ by $H\mu$, where $hH\equiv -1$ mod c, then the right-hand side of (51) is changed into the one with the factor ${(-1)}^{\frac{p-1}{2}}$ and with the new pair of parameters ${\alpha }^{\prime },{\beta }^{\prime }$. Hence

$$c{\left(2\pi cf\right)}^{s+p-1}{(-1)}^{\frac{p-1}{2}}\Gamma (1-p-s)g_{{\alpha }^{\prime },{\beta }^{\prime }}(s,x)=c{\left(2\pi cf\right)}^{-s}\Gamma \left(s\right)g_{\alpha ,\beta }(s,x),$$

which is (45).

Shifting the integration path in (42) to $\sigma =1-p-\gamma$ and applying (8) establishes the assertion. The residual function (44) may be found on [18][p. 48].

The degenerate case of (48) leads to a generalization of Rademacher’s functional equation. Indeed, (48) with $f=1$, $\gamma (0,0)$ reads

$$\begin{array}{cc}\hfill & c{\left(2\pi c\right)}^{-s}\Gamma \left(s\right)f_{0,0}(s,x)\hfill \\ \hfill & =\sum _{\lambda ,\nu =1}^c(\sum _{\mu =1}^c{e}^{\frac{2\pi ih\mu \nu }{c}}cos\frac{2\pi \lambda \mu }{c}\frac{1}{cos\frac{\pi }{2}s}\zeta \left(1-s,\frac{\lambda }{c}\right)\zeta \left(p+s,\frac{\nu }{c}\right)\hfill \\ \hfill & +\sum _{\mu =1}^c{e}^{\frac{2\pi ih\mu \nu }{c}}sin\frac{2\pi \lambda \mu }{c}\frac{1}{sin\frac{\pi }{2}s}\zeta \left(1-s,\frac{\lambda }{c}\right)\zeta \left(p+s,\frac{\nu }{c}\right)).\hfill \end{array}$$

(52)

Substituting (15) in (52) proves Rademacher-Apostol case [36]:

$${\left(2\pi c\right)}^{-s-\frac{p-1}{2}}\Gamma \left(s\right)Z_p(s,h)={\left(2\pi c\right)}^{s+\frac{p-1}{2}}{(-1)}^{\frac{p-1}{2}}\Gamma (-s)Z_p(1-p-s,H),$$

(53)

where

$$Z_p(s,h)=\sum _{\mu ,\nu =1}^c{e}^{\frac{2\pi ih\mu \nu }{c}}\zeta \left(s,\frac{\mu }{c}\right)\zeta \left(s+p,\frac{\nu }{c}\right).$$

(53) reduces to (8) for $p=1$. □

Other papers dealing with generalizations of the eta-function use

$$\mathcal{E}(s,h)=\sum _{\mu ,\nu =1}^c{e}^{\frac{2\pi ih\mu \nu }{c}}\zeta \left(s,\frac{\mu }{c}\right){\ell }_{s+1}\left(\frac{\nu }{c}\right).$$

instead of (6) and are feasible for description in the form of the Hecke correspondence. We hope to return to the study of this aspect and more general Dedekind sums including one with Kubert functions elsewhere. But we shall mentions one type of Estermann type in the nect section.

5. The Schoenberg Case

This section is concerned with [20], which is reproduced in [21][pp. 184-202, Chapter VIII]. On [21][p. 184] it is stated that the transition is made from Hecke’s Eisenstein series of weight $-2$[21][p. 164] to a linearly equivalent system containing non-analytic function $G_2$.

We stick to [20][p. 5 ], which is directly related to (5).

In particular,

$$\begin{array}{cc}\hfill & \zeta (s,\alpha ){\ell }_{s+1}\left(\beta \right)=\frac{-i{\left(2\pi \right)}^{2s}}{sin\pi s}(-e^{-\pi is}\zeta (-s,1-\beta ){\ell }_{1-s}\left(\alpha \right)\hfill \\ \hfill & +e^{\pi is}\zeta (-s,\beta ){\ell }_{1-s}(1-\alpha )+\zeta (-s,1-\beta ){\ell }_{1-s}(1-\alpha )\hfill \\ \hfill & -\zeta (-s,\beta ){\ell }_{1-s}\left(\alpha \right)).\hfill \end{array}$$

(54)

We write $\xi =e^{2\pi i\beta }$ and define the Lambert series [20][(20)]

$$U(x;\alpha ,\beta )=\sum _{\stackrel{n>0}{m>-\alpha }}\frac{{\xi }^n}{n}{e}^{-(m+\alpha )nx},x>0.$$

(55)

Then [20][(26)] considered the gamma transform of the Estermann type zeta function

$$U(x;\alpha ,\beta )=\frac{1}{2\pi i}{\int }_{\left(\varkappa \right)}\Gamma \left(s\right)\zeta (s,\alpha ){\ell }_{s+1}\left(\beta \right)c^{-x}\mathrm{d}s,$$

(56)

where $\varkappa >1$. If we substitute (54) into (56), then the integral is hardly tractable. This is why Schoenberg deduced only an asymptotuc formula for $U(x;\alpha ,\beta )$.

Let

$$\mathbf{a}=(a_1,a_2)\in {\mathbb{Z}}^2,\alpha =\alpha \left(\mathbf{a}\right)=\frac{a_1}{cN}+\frac{r}{c},\beta =\beta \left(\mathbf{a}\right)={\xi }_r,$$

(57)

where

$${\xi }_r=e^{2\pi i\left(\frac{a_1^{\prime }}{cN}+\frac{qr}{c}\right)},a_1^{\prime }=a{a}_1+c{a}_2.$$

(58)

Then we consider

$$X\left(\mathbf{a}\right)=X(a_1,a_2)=U(x;\alpha ,\beta )=U\left(2\pi cx;\frac{a_1}{cN}+\frac{r}{c},{\xi }_r\right).$$

(59)

But what is needed eventually is an expression for the even part $X(a_1,a_2)+X(-a_1,-a_2)$ ([20, p. 8]) and we prove the following theorm for the zeta-function of the even part.

Theorem 4.

For

$$Z(s,\alpha ,\beta )=\zeta (s,\alpha ){\ell }_{s+1}\left(\beta \right)+\zeta (s,1-\alpha ){\ell }_{s+1}(1-\beta )$$

and

$$\tilde{Z}(s,\alpha ,\beta )=\zeta (s,1-\beta ){\ell }_{s+1}\left(\alpha \right)+\zeta (-s,\beta ){\ell }_{1-s}(1-\alpha )$$

the functional equation

$$Z(s,\alpha ,\beta )=2{\left(2\pi \right)}^{2s}\tilde{Z}(-s,\alpha ,\beta )$$

(60)

holds.

Proof. 

On [20][p. 7], Schoenberg defined

$${\xi }_r^{\prime }=e^{2\pi i\left(-\frac{-a_1^{\prime }}{cN}+\frac{qr}{c}\right)}$$

(61)

and noted

$${\xi }_r^{\prime }={\xi }_{c-r}^{-1},$$

(62)

Hence

$$\alpha (-\mathbf{a})=1-\alpha \left(\mathbf{a}\right),\beta (-\mathbf{a})=1-\beta \left(\mathit{a}\right).$$

$$X(-\mathit{a})=U(x;1-\alpha ,1-\beta ).$$

(63)

It follows that when substituting from (54) in $X\left(\mathbf{a}\right)+X(-\mathbf{a})$, the sums with the third and the fourth terms vanish and we sum only first two terms of (54) and the sine function cancels. Hence the zeta-function $Z(s,\alpha ,\beta )$ of $X\left(\mathbf{a}\right)+X(-\mathbf{a})$ is

$$\begin{array}{cc}\hfill & Z(s,\alpha ,\beta )\hfill \\ \hfill & =\frac{-i{\left(2\pi \right)}^{2s}}{sin\pi s}\left(-e^{-\pi is}+e^{\pi is}\right)\hfill \\ \hfill & \left(\zeta (-s,1-\alpha \left(\mathbf{a}\right)){\ell }_{1-s}(-\alpha \left(\mathbf{a}\right))+\zeta (-s,\beta \left(\mathbf{a}\right)){\ell }_{1-s}\left(\alpha \left(\mathbf{a}\right)\right)\right)\hfill \\ \hfill & =2{\left(2\pi \right)}^{2s}\left(\zeta (-s,1-\beta ){\ell }_{1-s}\left(\alpha \right)+\zeta (-s,\beta ){\ell }_{1-s}(1-\alpha )\right),\hfill \end{array}$$

(64)

which proves (60). □

Hence what comes out is the Hecke gamma transform of a tractable function and the process onwards is verbatim to that of the preceding sections and we do not go into details.