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

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 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 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 $\left({*}_s\right)$: 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 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 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

(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 or as the case may be to deduce

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.

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

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

[34] deals with a generalization (38) of the eta-function which depends on the Hecke gamma transform of the zeta-function where $a,b$ are natural numbers, $(a,b)=1$. $Z_{a.b}\left(s\right)$ satisfies the Hecke functional equation Krätzel’s method is essentially that of Rademacher although he does not refer to [16] and we give a brief account on this point.

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

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

Other papers dealing with generalizations of the eta-function use 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.

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,

We write $\xi =e^{2\pi i\beta }$ and define the Lambert series [20][(20)] Then [20][(26)] considered the gamma transform of the Estermann type zeta function 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 where

Then we consider

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.