Unification of Chowla’s Problem and Maillet–Demyanenko Determinants

Abstract

Chowla’s (inverse) problem (CP) is to mean a proof of linear independence of cotangent-like values from non-vanishing of $L(1,\chi )$ $={\sum }_{n=1}^{\infty }\frac{\chi \left(n\right)}{n}$. On the other hand, we refer to determinant expressions for the (relative) class number of a cyclotomic field as the Maillet–Demyanenko determinants (MD). Our aim is to develop the theory of discrete Fourier transforms (DFT) with parity and to unify Chowla’s problem and Maillet–Demyanenko determinants (CPMD) as different-looking expressions of the relative class number via the Dedekind determinant and the base change formula.

1. Introduction and Main Results

Let $\chi$ be a Dirichlet character and let $L(s,\chi )={\sum }_{n=1}^{\infty }\frac{\chi \left(n\right)}{n^s}$ the Dirichlet L-function (defined for $\sigma =\mathrm{Re}s>1$ in the first place and then continued meromorphically by the functional equation). Then for $\chi$ non-principal, the series for $L(s,\chi )$ is convergent for $\sigma >0$ and the value $L(1,\chi )={\sum }_{n=1}^{\infty }\frac{\chi \left(n\right)}{n}$ is meaningful, which is one of the most important values in whole number theory. Indeed, Dirichlet’s aim was to show the non-vanishing of $L(1,\chi )$ (whence the infinitude of primes in an arithmetic progression) by that of the class number, thus leading to the most remarkable class number formula.

Chowla’s (original) problem [1,2,3] refer to an elementary proof of $L(1,\chi )\ne 0$ for odd $\chi$ through an elementary proof of the linear independence of cotangent values or similar stuff stated in Theorem 7. However, now, the inverse problem of Chowla is referred to as Chowla’s problem, which we also adopt, i.e., to prove linear independence of circular functions at rational arguments from the nonvanishing of $L(1,\chi )$.

On the other hand, Maillet(-Demyanenko) determinants refer to a determinant expression for the relative class number of a cyclotomic field, taken up by [4]. However, here expressions involving $L(k,\chi )$ are meaningful, and we state this case partially in the proof in Section 3. In early studies on the Maillet determinants, the entries were expressed in terms of the least non-negative residue $\overline{a}$ of a mod M, $a\equiv \overline{a}modM$, $0\le \overline{a}<M$, i.e., as in the first equality in (94). Only after the expression

$$\overline{a}=M\left({\overline{B}}_1\left(\frac{a}{M}\right)+\frac{1}{2}\right)$$

(1)

is introduced [5], has the situation become clearer. Cf. Section 4.1 for more details on CPMD.

Although there is an enormous amount of research on these two subjects, most of it is exclusively devoted to only one of them, save for Wang [6,7] who seems to be the only researcher that worked in both Chowla’s problem and the Maillet determinant. In ([7] pp. 306–307), Wang mentioned the link between them, but he did not go further.

Our aim in this paper is two-fold. First, we develop the theory of DFT further to prove the base change formula, Gauss’ first and second formula for the digamma function, and the pair of Eisenstein formulas. Second, as a very ample example, we unify Chowla’s problem and Maillet–Demyanenko determinants (CPMD) as different-looking expressions for the relative class number in Theorem 1, via the Dedekind determinant, the base change formula, and Euler product.

We shall thereby elucidate the underlying principles that make this merging possible. One of them is the notion of parity to be stated toward the end of this section. We develop DFT with parity and reveal to what extent we can accommodate Dirichlet L-functions as a special case of the Dirichlet series $D(s,f)$ with a periodic coefficient f of period M with orthogonality ${\sum }_{a=1}^{M}f\left(a\right)=0$. This corresponds to the case where the ramified functional equation reduces to an unramified one, i.e., the zeta-symmetry travels incognito with $D(s,f)$ specified to Dirichlet L-functions.

In the notation of Section 2.2 and Section 2.3, we state

Theorem 1. 

Chowla’s problem and Maillet–Demyanenko determinants amount to the different expressions for a relative class number on the grounds of the Dedekind determinant, Euler product, and the base change formulas

$$\begin{array}{cc}\hfill -\frac{\pi }{M}{\displaystyle \sum _{a=1}^M}\chi \left(a\right)cot\left(\pi \frac{a}{M}\right)& =L(1,\chi )=P(1,{\chi }^{*})L(1,{\chi }^{*})\hfill \\ \hfill & =i\frac{\pi \tau \left({\chi }^{*}\right)}{M}P(1,{\chi }^{*}){\displaystyle \sum _{a=1}^{M-1}}\overline{{\chi }^{*}}\left(a\right){\overline{B}}_1\left(\frac{a}{M}\right),\hfill \end{array}$$

(2)

where ${\chi }^{*}$ is the primitive character with conductor f inducing χ and $P(s,{\chi }^{*})$ is defined by (57).

The product overall odd primitive χ of (2) leads to a constant multiple of the relative class number (4).

Remark 1. 

Equation (2) (cf. Corollary 2) is the special case (for the Dirichlet L-function with odd χ) of the odd part of the base change formula (Theorem 4):

$$\begin{array}{cc}\hfill & \frac{\pi }{2M}{\displaystyle \sum _{a=1}^M}f\left(a\right)cot\left(\pi \frac{a}{M}\right)=-\frac{1}{M}{\displaystyle \sum _{a=1}^M}f\left(a\right){\psi }^o\left(\frac{a}{M}\right)=D(1,f^o)\hfill \\ \hfill & =\frac{1}{\sqrt{M}}{\displaystyle \sum _{a=1}^{M-1}}\widehat{f}\left(a\right){\ell }_1^o\left(\frac{a}{M}\right)=-i\frac{\pi }{\sqrt{M}}{\displaystyle \sum _{a=1}^{M-1}}\widehat{f}\left(a\right){\overline{B}}_1\left(\frac{a}{M}\right),\hfill \end{array}$$

(3)

where the exponent o indicates the odd part (5).

It turns out that DFT alone is not enough for the merging of CPMD, cf. Remark 3 below.

For Chowla’s problem, the odd part of $\psi$ is relevant, which is the cotangent function. This latter part follows from the functional equation or from the defining expression for $\psi$ and the partial fraction expansion for the cotangent function.

For the Maillet–Demyanenko determinants, $L(1,\chi )$ show up with imprimitive characters in general. They may be expressed in terms of the L-function with primitive character times the product arising from UFD property (Eratosthenes sieve). $L(1,\chi )$ with primitive odd $\chi$ has the expression in terms of the periodic Bernoulli polynomial, which leads to the expression for the relative class number of a cyclotomic field $\mathbb{Q}\left({\zeta }_M\right)$, with a primitive Mth root ${\zeta }_M$ of 1. In addition to (3), we need separability of the general Gauss sum and the Eratosthese sieve together with the product formula for the relative class number (odd part) $h_M^{-}$ as the quotient of the class number $h_M$ of the Mth cyclotomic field by the class number $h_M^{+}$ of its maximal real subfield (even part), i.e., multiplicative parity

$$h_M^{-}=\frac{h_M}{h_M^{+}}.$$

(4)

As we will see in Lemma 3, it is here that the Riemann–Bochner–Hecke (RHB) correspondence [8]—the zeta-symmetry—takes place implicitly to assure primitivity, which implies separability, cf. the third equality in (2). For RHB, whose equivalent expressions for the functional equation have been developed under the name of modular relations, compare [9].

In [10], parity in a wide sense was introduced, which means a classification of the whole into even and odd parts: W=E+O. This parity classification has been perceived in an implicit way in much mathematical literature and is partially formulated in [11,12,13], cf.

Table 1. Parity and ingredients.

objects	odd	even
character	odd	even
quadratic field	imaginary	real
function	Bernoulli polynomial	Clausen function
class number	relative class number	classumber of max. real subfield
determinant	Maillet	Maillet with Clausen function

, etc.

There are many objects which belong to the odd part, being more accessible than the even part. Imaginary quadratic fields, relative class number, Maillet–Demyanenko determinants (with periodic Bernoulli polynomials), linear independence of cotangent values (Chowla’s problem), Riemann’s posthumous fragment, etc. For the even part—Clausen functions, we refer to [5,14] etc.

We define the even resp. odd part by

$$\begin{array}{c}\hfill f^e\left(x\right)=\frac{1}{2}(f\left(x\right)+f(-x)),f^o\left(x\right)=\frac{1}{2}(f\left(x\right)-f(-x))\\ \hfill \end{array}$$

(5)

or in a similar way so that

$$\begin{array}{c}\hfill f=f^e+f^o.\end{array}$$

(6)

It follows that f is even $⟺f=f^e$ resp. f is odd $⟺f=f^o$.

2. Establishing the Underlying Principles

In this section we establish the Dedekind determinant Section 2.1, the base change formula in Section 2.2, and Euler product in Section 2.3.

2.1. Dedekind Determinant

Research on CPMD relies on the Dedekind determinant (relation), cf. [15], ([16] pp. 89–91), [17], which we state as

Lemma 1 

(Dedekind determinant). Let G be a finite Abelian group, N be a subgroup of G and $T\subset G$ be a complete system of representatives of $G/N$. For a character λ of N, let Δ be the set of all characters of G whose restriction to N is λ. Then for any complex-valued function f on G with

$$f\left(ah\right)=\lambda \left(h\right)f\left(a\right)(a\in G,h\in N),$$

(7)

we have

$$det{\left(f\left(a{b}^{-1}\right)\right)}_{a,b\in T}=\prod _{\chi \in \Delta }\left(\sum _{a\in T}\overline{\chi }\left(a\right)f\left(a\right)\right),$$

(8)

and if the principal character ${\chi }_0\in \Delta$, then

$$det{(f\left(a{b}^{-1}\right)-f\left(a\right))}_{\genfrac{}{}{0pt}{}{a,b\in T}{a,b\notin N}}=\prod _{\genfrac{}{}{0pt}{}{\chi \in \Delta }{\chi \ne {\chi }_0}}\left(\sum _{\genfrac{}{}{0pt}{}{a\in T}{a\notin N}}\overline{\chi }\left(a\right)f\left(a\right)\right).$$

(9)

For a more general notion of group characters, cf. ([18] pp. 30–34).

Lemma 1 is a special case of the following general theorem.

Theorem 2. 

Let G be a finite Abelian group written additively and $\widehat{G}$ its character group. Let ${\chi }_a$ be the characteristic function

$${\chi }_a\left(b\right)={\delta }_{ab}=\left\{\begin{array}{cc}1\hfill & a=b\hfill \\ 0\hfill & a\ne b\hfill \end{array}\right..$$

(10)

Then, for any function $f:G\to \mathbb{C}$, we have the base change formula

$$\sum _{a\in G}f\left(a\right){\chi }_a=f=\frac{1}{M}{\displaystyle \sum _{\epsilon \in \widehat{G}}^{}\sum _{a\in G}^{}}f\left(a\right)\epsilon (-a)\epsilon$$

(11)

and the relation between linear maps

$$\sum _{a\in G}f(a-b){\chi }_a=T{\chi }_b,T\epsilon =\sum _{a\in G}f\left(a\right)\epsilon (-a)\epsilon .$$

(12)

Their matrix expressions entail

$$det{\left(f(a-b)\right)}_{a,b}=\prod _{\epsilon \in \widehat{G}}\sum _{a\in G}\epsilon \left(a\right)f(-a).$$

(13)

We shall give a proof in the forthcoming paper in which we give a new establishment of the theory of DFT and even part of CPMD.

Proof of Lemma 1. 

Fix $\psi \in \Delta$. Then for any $f:G\to \mathbb{C}$ satisfying (7), we have

$$f{\psi }^{-1}:G/N\to \mathbb{C}.$$

For any $a\in N$, we have $f{\psi }^{-1}\left(a\right)=f\left(1a\right){\lambda }^{-1}\left(a\right)=1$. Hence,

$$\widehat{G/N}=\{\eta =\chi {\psi }^{-1}|\chi \in \Delta \}.$$

(14)

Writing $G/N=\{\overline{a}=amodN\}$, we apply Theorem 2 to obtain

$$\prod _{\eta \in \widehat{G/N}}{\displaystyle \sum _{\overline{a}\in G/N}^{}}\eta \left(\overline{a}\right)f{\psi }^{-1}\left({\overline{a}}^{-1}\right)=det(f{\psi }^{-1}{({\overline{a}}^{-1}\overline{b})}_{a,b\in T}).$$

(15)

By (14), the LHS of (15) is

$$\prod _{\chi \in \Delta }{\displaystyle \sum _{a\in T}^{}}\chi {\psi }^{-1}\left(a\right)f{\psi }^{-1}\left(a^{-1}\right)=\prod _{\chi \in \Delta }{\displaystyle \sum _{a\in T}^{}}\chi \left(a\right)f\left(a^{-1}\right),$$

which is the LHS of (8). The RHS is $det\left(Af\left(a^{-1}b\right)B\right),$ where $A=\left(\begin{array}{ccc}{\psi }^{-1}\left(a_1^{-1}\right)& & O\\ & \ddots & \\ O& & {\psi }^{-1}\left(a_{\left|G\right|}^{-1}\right)\end{array}\right)$, say and B is similar with ${\psi }^{-1}\left(a^{-1}\right)$ replaced by ${\psi }^{-1}\left(b\right)$. Hence

$$det\left(Af\left(a^{-1}b\right)B\right)=\prod _{a\in T}{\psi }^{-1}\left(a^{-1}\right)det\left(f\left(a^{-1}b\right)\right)\prod _{b\in T}{\psi }^{-1}\left(b\right)=det\left(f\left(a^{-1}b\right)\right),$$

(16)

which is the RHS of (8), thereby competing the proof. □

Remark 2. 

Applying Theorem 2 to $G=\mathbb{Z}/M\mathbb{Z}$, and by the theory of periodic functions of period M, it’s clear (11) leads to the base change relation (34). According to the labeling (19), the sum over $j=1,\cdots ,M$ is the one over characters $\epsilon \in \widehat{G}$.

Lemma 1, which is a consequence of Theorem 2, is applied to $G={(\mathbb{Z}/M\mathbb{Z})}^{\times }$ to handle the Maillet–Demyanenko determinants. In [5], a subgroup $H\not\ni 1$ is considered (which is to be the congruence group by class field theory), and N is taken to be $H\{\pm 1\}$. In our application, we take $H=1$ and $N=\{\pm 1\}$, in which case $G/N$ is the Galois group of $K/K^{+}$.

2.2. DFT and Periodic Dirichlet Series

This is the main body of the paper, and we develop the theory of DFT (discrete Fourier transform) [11,12,19,20,21] further to prove the base change formula, Gauss’ first and second formula for the digamma function, and the pair of Eisenstein formulas.

We also elucidate to what extent we can cover Dirichlet L-functions as a special case of the Dirichlet series $D(s,f)$ with periodic coefficient f of period M with orthogonality ${\sum }_{a=1}^{M}f\left(a\right)=0$.

The most relevant result is Theorem 4, which is an enhanced version of the result in [22] and entails (3).

Let $C\left(M\right)$ be the vector space of all periodic arithmetic functions f with period M:

$$C\left(M\right)=\{f:\mathbb{Z}\to \mathbb{C}|f(n+M)=f\left(n\right)\}.$$

(17)

The inner product of $f_1,f_2\in C\left(M\right)$ is defined by

$$\left(f_1,f_2\right)={\displaystyle \sum _{amodM}^{}}{f}_1\left(a\right)\overline{f_2\left(a\right)},$$

(18)

where the bar $\overline{·}$ means the complex conjugation of ·. $C\left(M\right)$ becomes an inner product space.

Let

$${\epsilon }_j\left(a\right)=e^{2\pi ija/M},1\le j\le M,$$

(19)

be the additive characters of $\mathbb{Z}/M\mathbb{Z}$, where a is an integer variable. Then

$$({\epsilon }_j,{\epsilon }_k)={\delta }_{jk}M,1\le j,k\le M,$$

(20)

so that $\mathcal{E}:=\left\{\frac{1}{\sqrt{M}}{\epsilon }_j\right|1\le j\le M\}$ is an orthonormal system (ONS). Suppose it is an orthogonal normal basis (ONB). Then the orthogonal expansion reads

$$f={\displaystyle \sum _{b=1}^M}\left(f,\frac{1}{\sqrt{M}}{\epsilon }_b\right)\frac{1}{\sqrt{M}}{\epsilon }_b={\displaystyle {\displaystyle \sum _{b=1}^M}}\widehat{f}\left(b\right)\frac{1}{\sqrt{M}}{\epsilon }_b$$

(21)

for every $f\in C\left(M\right)$. Thus the DFT $\widehat{f}$ (or the bth Fourier coefficient) of $f\in C\left(M\right)$ is to be defined by

$$\widehat{f}\left(b\right)=\left(f,\frac{1}{\sqrt{M}}{\epsilon }_b\right)=\frac{1}{\sqrt{M}}{\displaystyle \sum _{a=1}^M}{\epsilon }_b(-a)f\left(a\right).$$

(22)

and the Fourier inversion or Fourier expansion formula holds true:

$$f\left(a\right)=\frac{1}{\sqrt{M}}{\displaystyle \sum _{b=1}^M}\widehat{f}\left(b\right){\epsilon }_b\left(a\right)=\widehat{\widehat{f}}(-a).$$

(23)

Since (23) is the unique expression of f with respect to the ONS $\mathcal{E}$, the set $\mathcal{E}$ forms a basis (i.e., ONB) of $C\left(M\right)$ and a fortiori $C\left(M\right)$ is a vector space of dimension M.

By (20), (22) reads for $f={\epsilon }_j$

$${\widehat{\epsilon }}_j\left(b\right)=\frac{1}{\sqrt{M}}{\delta }_{jb}.$$

(24)

As in (5), we define the even resp. odd part of f: $f=f^e+f^o$. Then the parity inherits to the DFT. To find another natural basis, let ${\chi }_a$ be the characteristic function ${\chi }_a$ mod M ([23] p. 73).

$${\chi }_a\left(n\right)=\left\{\begin{array}{cc}1\hfill & n\equiv amodM\hfill \\ 0\hfill & n\not\equiv amodM\hfill \end{array}\right..$$

(25)

Then $\left\{{\chi }_a\right|1\le a\le M\}$ is a basis of $C\left(M\right)$ and

$$\sqrt{M}{\widehat{\chi }}_a\left(n\right)={\displaystyle \sum _{j=1}^M}{\epsilon }_n(-j){\chi }_a\left(j\right)={\epsilon }_a(-n).$$

(26)

For $f\in C\left(M\right)$, let

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

(27)

Since

$${\displaystyle \sum _{n=1}^{\infty }}\frac{\left|f\right(n\left)\right|}{n^{\sigma }}\ll \zeta \left(\sigma \right),$$

where $\zeta \left(s\right)$ is the Riemann zeta function (102), the series in (27) is absolutely convergent for $\sigma >1$. Let $D\left(M\right)$ denote the set of all Dirichlet series of the form (27):

$$D\left(M\right)=\left\{D\right(s,f\left)\right|f\in C\left(M\right)\}$$

(28)

for $\sigma >1$ in the first instance. Then it forms a vector space of dimension M canonically isomorphic to $C\left(M\right)$. One of the bases of $D\left(M\right)$ is $\left\{{\ell }_s\left(\frac{a}{M}\right)\right|1\le a\le M\}$, where ${\ell }_s\left(x\right)$ is the Lerch zeta function in (100). Hence, we have

$$D(s,f)=\frac{1}{\sqrt{M}}{\displaystyle \sum _{a=1}^M}\widehat{f}\left(a\right){\ell }_s\left(\frac{a}{M}\right)=\frac{1}{\sqrt{M}}{\displaystyle \sum _{a=1}^{M-1}}\widehat{f}\left(a\right){\ell }_s\left(\frac{a}{M}\right)+\frac{\widehat{f}\left(M\right)}{\sqrt{M}}\zeta \left(s\right).$$

(29)

It follows that $D(s,f)$ can be continued meromorphically over the whole plane and that it is entire if and only if $\widehat{f}\left(M\right)=0$, which reads with (22)

$$\widehat{f}\left(M\right)=\frac{1}{\sqrt{M}}{\displaystyle \sum _{a=1}^M}f\left(a\right)=0.$$

(30)

Another basis of $D\left(M\right)$ is $\left\{D(s,{\chi }_a)\right|1\le a\le M\}$, where

$$D(s,{\chi }_a)={\displaystyle \sum _{n=1}^{\infty }}\frac{{\chi }_a\left(n\right)}{n^s}={\displaystyle \sum _{\stackrel{n=1}{n\equiv amodM}}^{\infty }}\frac{1}{n^s}=\zeta (s,a,M)=M^{-s}\zeta \left(s,\frac{a}{M}\right),$$

(31)

where $\zeta (s,a,M)$ indicates the partial zeta function and

$$\left\{x\right\}=x-\left[x\right]={\overline{B}}_1\left(x\right)+\frac{1}{2}$$

(32)

is the fractional part of x, $\left[x\right]$ being the integral part of x. Note that it is $\zeta (s,1-\left\{\frac{a}{M}\right\})$ that belongs to $D\left(M\right)$ rather than $\zeta (s,\left\{\frac{a}{M}\right\})$, cf. Remark 2 and

Table 2. Vector spaces and their bases.

vector space	$C\left(M\right)$	$D\left(M\right)$
add. character basis	${\epsilon }_a$	${\ell }_s\left(\frac{a}{M}\right)$
characteristic. ftn. basis	${\chi }_a$	$\zeta \left(s,1-\left\{\frac{a}{M}\right\}\right)$

. Hence in parallel to (29), we have another expression

$$D(s,f)={\displaystyle \sum _{a=1}^M}f\left(a\right)\zeta \left(s,a,M\right)=\frac{1}{M^s}{\displaystyle \sum _{a=1}^M}f\left(a\right)\zeta \left(s,\frac{a}{M}\right).$$

(33)

Theorem 3. 

The space $D\left(M\right)$ of all Dirichlet series $D(s,f)$ with periodic coefficients $f\in C\left(M\right)$ of period M is an inner product space of dimension M canonically isomorphic to $C\left(M\right)$. $C\left(M\right)$ resp. $D\left(M\right)$ has ONBs (19) and (25) resp. ${\ell }_s\left(\frac{a}{M}\right)$ and $\zeta \left(s,1-\left\{\frac{a}{M}\right\}\right)$.

$D(s,{\widehat{f}}^o)$ is an entire function and so is $L(s,\chi )$ for non-principal Dirichlet character χ.

The following is a slightly enhanced version of ([22] Proposition 1.)

Theorem 4 

(Base change formula). For $f\in C\left(M\right)$,

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

(34)

which entails the formula for the Laurent constant

$$\begin{array}{cc}\hfill -\frac{1}{\sqrt{M}}\widehat{f}\left(M\right)-\frac{1}{M}{\displaystyle \sum _{k=1}^M}f\left(k\right)\psi \left(\frac{k}{M}\right)& =\underset{s\to 1}{lim}\left(D(s,f)-\frac{\widehat{f}\left(M\right)}{\sqrt{M}}\zeta \left(s\right)\right)\hfill \\ \hfill & =\frac{1}{\sqrt{M}}{\displaystyle \sum _{k=1}^{M-1}}\widehat{f}\left(k\right){\ell }_1\left(\frac{k}{M}\right)+\frac{\widehat{f}\left(M\right)}{\sqrt{M}}\gamma \hfill \end{array}$$

(35)

as well as the odd part

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

(36)

say. (35) with $\widehat{f}\left(M\right)=0$ (esp. for odd f) is a special case of (36) as $s\to 1$.

Theorem 5. 

Base change Formula (34) amounts to the pair of generalizations of Eisenstein formula with $f={\epsilon }_a$ resp. $f={\chi }_a$ resp. for which (24) resp. (26) holds:

$${\displaystyle \sum _{a=1}^{M-1}}{\epsilon }_a(-b){\ell }_s\left(\frac{a}{M}\right)=M^{1-s}\zeta \left(s,\frac{b}{M}\right)-\zeta \left(s\right),$$

(37)

resp.

$${\displaystyle \sum _{a=1}^M}{\epsilon }_b(-a)\zeta \left(s,\frac{a}{M}\right)=M^s{\ell }_1\left(\frac{b}{M}\right).$$

(38)

(37) entails Gauss’ first formula for the digamma function at rational argument

$$\begin{array}{cc}\hfill \psi \left(\frac{a}{M}\right)& =-\gamma -logM+\pi i{\displaystyle \sum _{b=1}^{M-1}}{\epsilon }_a(-b){\overline{B}}_1\left(\frac{b}{M}\right)-{\displaystyle \sum _{j=1}^{M-1}}{\epsilon }_a\left(j\right)A_1\left(\frac{j}{M}\right)\hfill \\ \hfill & =-\gamma -logM-\frac{\pi }{2}cot\frac{a}{M}\pi +{\displaystyle \sum _{j=1}^{M-1}}cos\frac{2\pi aj}{M}logsin2\frac{\pi j}{M}\hfill \end{array}$$

(39)

while (38) entails Gauss’ second formula ($b\not\equiv 0$ mod M)

$$\begin{array}{cc}\hfill \frac{1}{M}{\displaystyle \sum _{a=1}^M}{\epsilon }_a(-b)\psi \left(\frac{a}{M}\right)={\ell }_1\left(\frac{b}{M}\right)& =-log\left(1-e^{2\pi i\frac{b}{M}}\right)\hfill \\ \hfill & =A_1\left(\frac{b}{M}\right)-\pi i{\overline{B}}_1\left(\frac{b}{M}\right)\hfill \end{array}$$

(40)

and the Kubert identity

$$M^{-1}{\displaystyle \sum _{r=0}^{M-1}}\psi \left(\frac{x+r}{M}\right)=\psi \left(x\right)-logM$$

with $x=1$ ($b\equiv 0$ mod M).

Gauss’ first formula resp. Gauss’ second formula is also a consequence of (35) with $f={\chi }_a$ resp. $f={\epsilon }_a$.

Proof. 

(39) resp. (40) follows from the Laurent constant of (37) at $s=1$ resp. from separating the cases $b\equiv M$ or not in (38).

To derive Gauss’ formulas from (35), we argue as follows. Substituting (114) and (115) in (35), we deduce that

$$\psi \left(\frac{a}{M}\right)=-\gamma -logM+\pi i{\displaystyle \sum _{b=1}^{M-1}}{\epsilon }_a(-b){\overline{B}}_1\left(\frac{b}{M}\right)-{\displaystyle \sum _{j=1}^{M-1}}cos\frac{2\pi aj}{M}{A}_1\left(\frac{j}{M}\right).$$

(41)

Replacing the ${\widehat{B}}_1$ term with the reverse Eisenstein formula (45), (39) follows. (40) follows with the choice $f={\epsilon }_a$. □

It follows that

$$D(1,f)=\frac{\pi }{2M}{\displaystyle \sum _{a=1}^{M-1}}f\left(a\right)cot\frac{a}{M}\pi$$

(42)

for f odd and

$$D(1,f)=-\frac{1}{\sqrt{M}}{\displaystyle \sum _{a=1}^{M-1}}\widehat{f}\left(a\right)log\left(2sin\frac{a}{M}\pi \right)$$

(43)

for f even if $\widehat{f}\left(M\right)=0$.

It is known [24] that if (42) and (43) hold for Dirichlet characters, then (39) is equivalent to these finite expressions.

Corollary 1. 

The pair of the Eisenstein formula ([25])

$$\frac{1}{\sqrt{M}}\widehat{cot}\left(\frac{a}{M}\right)={\displaystyle \sum _{b=1}^{M-1}}{\epsilon }_a(-b)cot\left(\frac{b}{M}\right)=2i{\overline{B}}_1\left(\frac{a}{M}\right)$$

(44)

resp. the (reverse) Eisenstein formula

$$\sqrt{M}{\widehat{\overline{B}}}_1\left(\frac{a}{M}\right)={\displaystyle \sum _{b=1}^{M-1}}{\epsilon }_a(-b){\overline{B}}_1\left(\frac{b}{M}\right)=-\frac{1}{2i}cot\frac{\pi a}{M}$$

(45)

is the odd part of Theorem 5, i.e., a consequence of

$$\begin{array}{c}\hfill \frac{\pi }{M}{\displaystyle \sum _{b=1}^{M-1}}f\left(b\right)cot\left(\frac{\pi b}{M}\right)=\frac{-2\pi i}{\sqrt{M}}{\displaystyle \sum _{b=1}^{M-1}}\widehat{f}\left(b\right){\overline{B}}_1\left(\frac{b}{M}\right)\end{array}$$

(46)

with $f={\epsilon }_a$ and (24) resp. $f={\chi }_a$ and (26).

In [26] and references therein, a generalization of the Eisenstein formula

$${\displaystyle \sum _{j=1}^{M-1}}{e}^{-2\pi i\frac{a}{M}j}{l}_s\left(\frac{j}{M}\right)=M^{1-s}\zeta \left(s,\frac{a}{M}\right)-\zeta \left(s\right),$$

(47)

has been used, it being understood that for $s=1$, the right-hand side means the limit as $s\to 1$. This follows from (34) with $f={\chi }_a$ and also follows from the odd part of (34)

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

(48)

Corollary 2. 

Let χ be a non-principal Dirichlet character mod M. Then

$$\frac{1}{M^s}{\displaystyle \sum _{a=1}^{M-1}}\chi \left(a\right)\zeta \left(s,\frac{a}{M}\right)=L(s,\chi )=\frac{1}{\sqrt{M}}{\displaystyle \sum _{a=1}^{M-1}}\widehat{\chi }\left(a\right){\ell }_s\left(\frac{a}{M}\right),$$

(49)

which entails

$$\frac{1}{M}{\displaystyle \sum _{a=1}^{M-1}}\chi \left(a\right)\psi \left(\frac{a}{M}\right)=L(1,\chi )=\frac{1}{\sqrt{M}}{\displaystyle \sum _{a=1}^{M-1}}\widehat{\chi }\left(a\right){\ell }_1\left(\frac{a}{M}\right).$$

(50)

Theorem 6 

(Refs. [11,12]). Suppose f is a periodic arithmetic function with period M. Then

$$D(1-s,f)={\left(\frac{\pi }{M}\right)}^{\frac{1}{2}-s}\left(\frac{\Gamma \left(\frac{s}{2}\right)}{\Gamma \left(\frac{1-s}{2}\right)}D(s,{\widehat{f}}^e)+\frac{\Gamma \left(\frac{1+s}{2}\right)}{\Gamma \left(1-\frac{s}{2}\right)}D(s,{\widehat{f}}^o)\right),$$

(51)

or the ramified functional equation

$$\begin{array}{cc}\hfill & \Gamma \left(\frac{s}{2}\right)\Gamma \left(\frac{1+s}{2}\right){\left(\frac{\pi }{M}\right)}^{-\frac{s}{2}}D(s,f)\hfill \\ \hfill & =\Gamma \left(\frac{1-s}{2}\right)\Gamma \left(\frac{1+s}{2}\right){\left(\frac{\pi }{M}\right)}^{-\frac{1-s}{2}}D(1-s,{\widehat{f}}^e)\hfill \\ \hfill & +\Gamma \left(\frac{s}{2}\right)\Gamma \left(1-\frac{s}{2}\right){\left(\frac{\pi }{M}\right)}^{-\frac{1-s}{2}}D(1-s,{\widehat{f}}^o).\hfill \end{array}$$

(52)

This is equivalent to the generalized Euler identity

$$\tilde{D}(1-s,f)={\sqrt{\pi }}^{-1}{2}^{1-s}\Gamma \left(s\right)\left(cos\left(\frac{\pi s}{2}\right)\tilde{D}(s,{\widehat{f}}^e)+sin\left(\frac{\pi s}{2}\right)\tilde{D}(s,{\widehat{f}}^o)\right),$$

(53)

where

$$\tilde{D}(s,f)={\left(\frac{\pi }{M}\right)}^{-\frac{s}{2}}D(s,f).$$

(54)

(53) above is another statement.

Proof. 

Recall Equations (51) and (52), i.e., the ramified functional equation, on clearing the denominators and multiplying by ${\left(\frac{\pi }{M}\right)}^{\frac{s-1}{2}}$. □

This theorem has been elucidated in the light of the ramified functional equation [10].

2.3. Dirichlet Characters and L-Functions

The aim of this subsection is to establish the second and the third equality in (2) of our main theorem, i.e., the treatment of imprimitive characters and associated L-functions.

We introduce Dirichlet characters $\chi$ resp. associated Dirichlet L functions $L(s,\chi )=D(s,\chi )$ as important members of $C\left(M\right)$ resp. $D\left(M\right)$, cf. [19,27,28], ([21] Chapter 3), etc. Along with the additive characters ${\epsilon }_j$ in (19), one considers characters of the multiplicative group ${(\mathbb{Z}/M\mathbb{Z})}^{\times }$ of residue classes mod M, an Abelian group of order $\phi \left(M\right)$, the Euler function counting the number of integers $1,\cdots ,M$ relatively prime to M. From each reduced residue class character $\chi \in \widehat{{(\mathbb{Z}/M\mathbb{Z})}^{\times }}$ there arises its 0-extension denoted by the same symbol called a Dirichlet character mod M. The 0-extension of the trivial character is called the principal character and denoted ${\chi }_0$. For a non-principal character $\chi$ there exists the least period $f|M$ (f is a divisor of M) and the unique Dirichlet character ${\chi }^{*}$ to the modulus f such that

$$\chi \left(n\right)={\chi }_0{\chi }^{*}\left(n\right)=\left\{\begin{array}{cc}{\chi }^{*}\left(n\right)\hfill & (n,M)=1\hfill \\ 0\hfill & (n,M)>1,\hfill \end{array}\right.$$

(55)

f is called the conductor of $\chi$ and $\chi$ is said to be induced by the primitive character. A Dirichlet character mod M is primitive if and only if its conductor is M. The Dirichlet series (27) with Dirichlet character coefficients is called the Dirichlet L-functions and denoted

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

(56)

for $\sigma >1$ in the first instance. It is continued meromorphically over the whole plane with a possible simple pole at $s=1$. Many formulas for L-functions take their simplest forms only for primitive charactersFor the case of Maillet–Demyanenko determinants, imprimitive characters appear, and the following well-known lemma is essential.

Lemma 2. 

If χ is a non-principal character mod M induced by the primitive character ${\chi }^{*}$, then

$$L(s,\chi )=L(s,{\chi }^{*})P(s,{\chi }^{*}),P(s,{\chi }^{*})=\prod _{p|M}\left(1-\frac{{\chi }^{*}\left(p\right)}{p^s}\right).$$

(57)

If χ is a primitive character mod M, then $P(s,{\chi }^{*})=1$.

Proof of (57) given on ([21] p. 101) depends on the relative primality principle on ([21] p. 83), which in turn is based on the Möbius inversion formula.

Another proof of (57) may be given by the multiplicative Eratosthenes sieve on ([21] pp. 93–94). Let $P=\{p_1\cdots p_k\}$ denote the set of all prime factors of M that do not divide f. Then

$$\begin{array}{cc}\hfill & {\left(1-\frac{{\chi }^{*}\left(p_1\right)}{p_1^s}\right)}^{-1}L(s,\chi )=\prod _{p\ne p_1}{\left(1-\frac{\chi \left(p\right)}{p^s}\right)}^{-1},\hfill \\ \hfill & \cdots \hfill \\ \hfill & {\prod _{p\in P}\left(1-\frac{{\chi }^{*}\left(p\right)}{p^s}\right)}^{-1}L(s,\chi )=\prod _{p\notin P}{\left(1-\frac{\chi \left(p\right)}{p^s}\right)}^{-1}=L(s,{\chi }^{*}),\hfill \end{array}$$

(58)

whence (57).

To apply Corollary 2, we need to consider the DFT (22) of $\chi$, which is known as the general Gauss sum

$$\tau (\chi ,n)={\displaystyle \sum _{a=1}^M}\chi \left(a\right)e^{\frac{2\pi ina}{M}}=\sqrt{M}\widehat{\chi }(-n).$$

(59)

The (normalized) Gauss sum

$$\tau \left(\chi \right)=\tau (\chi ,1)={\displaystyle \sum _{a=1}^M}\chi \left(a\right)e^{\frac{2\pi ia}{M}}$$

(60)

plays an essential role.

Lemma 3. 

(i) Suppose χ is a non-principal character mod M. Then the general Gauss sum (59) is separable, i.e.,

$$\tau (\chi ,n)=\overline{\chi }\left(n\right)\tau \left(\chi \right)$$

(61)

if and only if χ is a primitive character mod M.

(ii) A non-principal character χ mod M is primitive if and only if the functional equation holds true, which reads in the form of a generalized Euler identity

$$\begin{array}{cc}\hfill & L(s,\chi )=\frac{\tau \left(\chi \right)}{\sqrt{M}}\frac{1}{\pi }{\left(2\pi \right)}^s\Gamma (1-s)\times \hfill \\ \hfill & \left(sin\left(\frac{\pi s}{2}\right)(1-\mathfrak{a}\left(\chi \right))L(1-s,\overline{\chi })+icos\left(\frac{\pi s}{2}\right)\mathfrak{a}\left(\chi \right)L(1-s,\overline{\chi })\right),\hfill \\ \hfill \end{array}$$

(62)

where $\mathfrak{a}\left(\chi \right)$ is the parity symbol

$$\mathfrak{a}=\mathfrak{a}\left(\chi \right)=\frac{1-\chi (-1)}{2}=\left\{\begin{array}{cc}0\hfill & \chi (-1)=1\hfill \\ 1\hfill & \chi (-1)=-1\hfill \end{array}\right..$$

(63)

(i) is due to ([27] Theorem 8.19, p. 171) ([21] Theorem 5.4, p.131) and (ii) is due to ([21] Exercise 72, p. 132).

The proper form of the functional equation of $L(s,\chi )$ for an odd imprimitive $\chi$ induced by the primitive ${\chi }^{*}$ seems to be

$$P(1-s,{\overline{\chi }}^{*})L(s,\chi )=\frac{i\tau \left(\chi \right)}{\pi \sqrt{M}}{\left(2\pi \right)}^s\Gamma (1-s)cos\left(\frac{\pi s}{2}\right)P(s,{\chi }^{*})L(1-s,\overline{\chi }).$$

(64)

It follows from (59) and (61) that

$$\widehat{\chi }\left(n\right)=\frac{\tau \left(\chi \right)}{\sqrt{M}}\chi (-1)\overline{\chi }\left(n\right)$$

(65)

and for a primitive $\chi$

$$\tau \left(\chi \right)\tau \left(\overline{\chi }\right)={\left|\tau \left(\chi \right)\right|}^2=M.$$

(66)

The generalized Bernoulli numbers

$$B_{k,\chi }=f^{k-1}{\displaystyle \sum _{a=1}^{f-1}}\chi \left(a\right)B_k\left(\frac{a}{f}\right)$$

(67)

have been introduced for a primitive $\chi$ with conductor f to express the value $L(1-k,\chi )$:

$$L(1-k,\chi )=-\frac{B_{k,\chi }}{k}.$$

(68)

Cf. (90), ([17] pp. 30–31, 37).

(Ref. [29] p. 12) gives for a primitive $\chi$ with conductor f and positive integers $k\equiv \mathfrak{a}$ mod 2

$$L(k,\chi )={(-1)}^{\frac{k-\mathfrak{a}}{2}}\frac{\tau \left(\chi \right)}{2i^{\mathfrak{a}}}{\left(\frac{2\pi }{f}\right)}^k\frac{1}{\Gamma \left(k\right)}L(1-k,\overline{\chi })$$

(69)

and

$$L(k,\chi )={(-1)}^{1+\frac{k-\mathfrak{a}}{2}}\frac{\tau \left(\chi \right)}{2i^{\mathfrak{a}}}{\left(\frac{2\pi }{f}\right)}^k\frac{1}{k!}{B}_{k,\overline{\chi }},$$

(70)

Both are special cases of the functional equation. In particular, for a primitive odd $\chi$

$$L(1,\chi )=-\pi i\frac{\tau \left(\chi \right)}{f}L(0,\overline{\chi })=\pi i\frac{\tau \left(\chi \right)}{f}{B}_{1,\overline{\chi }}.$$

(71)

Lemma 4. 

Let $\chi \ne {\chi }_0$ be a Dirichlet character mod M with conductor $f=f_{\chi }$ and let $P(s,{\chi }^{*})$ be the product in (57). Then for $k\in \mathbb{N}$,

$$B_{k,\chi }=M^{k-1}{\displaystyle \sum _{a=1}^{M-1}}\chi \left(a\right)B_k\left(\frac{a}{M}\right)=B_{k,{\chi }^{*}}P(1-k,{\chi }^{*}).$$

(72)

In particular,

$$B_{1,\chi }={\displaystyle \sum _{a=1}^{M-1}}\chi \left(a\right){\overline{B}}_1\left(\frac{a}{M}\right)=B_{1,{\chi }^{*}}P(0,{\chi }^{*})=-i\frac{f}{\pi \tau \left({\chi }^{*}\right)}L(1,{\chi }^{*})P(0,{\chi }^{*}).$$

(73)

(72) is contained in ([5] Lemma 2) and can be proved by the same Eratosthenes sieve that is used to prove (57).

It is known that $0\ne f\in C\left(M\right)$ is a Dirichlet character mod M if it is multiplicative and $f\left(n\right)=0$ for $(n,M)>1$ and that Dirichlet characters $\chi$ mod M satisfy the orthogonality.

It follows that a non-principal Dirichlet character is a multiplicative function $\in C\left(M\right)$, which is a 0-extension with orthogonality. Thus we apply (72), which in the long run depends on the primitivity of the associated inducing character and which is equivalent to the unramified functions equation, to arrive at (82) in the proof of the main theorem.

3. Proof of the Main Theorem

Proof of Theorem 1. 

In both cases, we apply Lemma 1 with $G={(\mathbb{Z}/M\mathbb{Z})}^{\times }$ the group of reduced residue classes mod M, $N=\{\pm 1\}$, (74) as a complete system T of representatives of $G/N$ and $\Delta =X^o$ the set of all odd Dirichlet characters.

To prove the Chowla problem part of Theorem 1, it suffices to prove one assertion in Theorem 7. We modify the proof of Shirasaka [30] a little and we may choose as T the complete set of representatives of $G/\{\pm 1\}$

$$T=\left\{a|1\le a<\frac{M}{2},(a,M)=1\right\},$$

(74)

where $(a,M)$ indicates the greatest common divisor of a and M once and for all, so that $(a,M)=1$ means that a and M are relatively prime. Suppose ($c_a\in \mathbb{Q}$)

$$0=\sum _{a\in T}{c}_{a}cot\frac{\pi a}{M}=i\sum _{a\in T}{c}_a\xi \left(a\right),$$

(75)

say, where

$$\xi \left(a\right)=-icot\frac{\pi a}{M}=\frac{{\zeta }^{a/2}+{\zeta }^{-a/2}}{{\zeta }^{a/2}-{\zeta }^{-a/2}}$$

(76)

and $\zeta =e^{\frac{2\pi i}{M}}$.

To accommodate this in the framework of Lemma 1, we rewrite (75) as

$$0=\sum _{a\in T}{c}_a\xi \left(a{b}^{-1}\right),$$

(77)

by applying the automorphism ${\sigma }_b^{-1}$ in (97) and consider the determinant $det{\left(\xi \left(a{b}^{-1}\right)\right)}_{a,b\in T}$. For an odd $\psi$ we have

$$det{\left(\xi \left(a{b}^{-1}\right)\right)}_{a,b\in T}=det{\left(\psi \left(a{b}^{-1}\right)\xi \left(a{b}^{-1}\right)\right)}_{a,b\in T}.$$

(78)

Indeed, the right-hand side we may factor out the coefficient ${\prod }_{a\in T}\psi \left(a\right)\times {\prod }_{b\in T}\overline{\psi }\left(b\right)$, which is 1.

Now apply (8) with $f\left(a\right)=\psi \left(a\right)\xi \left(a\right)$ which is even, to deduce that

$$\begin{array}{cc}\hfill det{\left(\psi \left(a{b}^{-1}\right)\xi \left(a{b}^{-1}\right)\right)}_{a,b\in T}& =\prod _{\chi \in X^e}\left(\sum _{a\in T}\overline{\chi }\left(a\right)\psi \left(a\right)\xi \left(a\right)\right)\hfill \\ \hfill & =\prod _{\chi \in X^e}\left(\sum _{a\in T}\overline{\chi }\psi \left(a\right)\xi \left(a\right)\right),\hfill \end{array}$$

so that

$$det{\left(\xi \left(a{b}^{-1}\right)\right)}_{a,b\in T}=\prod _{\chi \in X^o}\left(\sum _{a\in T}\chi \left(a\right)\xi \left(a\right)\right)$$

(79)

in view of (78). We may express $\xi \left(a\right)$ as

$$\xi \left(a\right)=-\frac{2iM}{\pi \phi \left(M\right)}{\displaystyle \sum _{\psi \in X^o}^{}}\psi \left(a\right)L(1,\overline{\psi }).$$

(80)

From the first equality of (2) by orthogonality. Substituting (80) in (79), we compute each factor

$$\sum _{a\in T}\chi \left(a\right)\xi \left(a\right)=-\frac{2iM}{\pi \phi \left(M\right)}{\displaystyle \sum _{\psi \in X^o}^{}}L(1,\overline{\psi })\sum _{a\in T}\chi \left(a\right)\psi \left(a\right).$$

Rewriting the inner sum ${\sum }_{a\in T}\chi \left(a\right)\psi \left(a\right)$ as

$$\frac{1}{2}\sum _{a\in T\cup (-T)}\chi \left(a\right)\psi \left(a\right)=\frac{1}{2}{\displaystyle \sum _{a=1}^M}\chi \psi \left(a\right),$$

which is $\frac{1}{2}\phi \left(M\right)$ only when $\psi =\overline{\chi }$ by orthogonality. Hence (79) amounts to

$$det{\left(\xi \left(a{b}^{-1}\right)\right)}_{a,b\in T}=\prod _{\chi \in X^o}-\frac{iM}{\pi }L(1,\chi ).$$

(81)

Appealing to the non-vanishingness of $L(1,\chi )$, we conclude that the determinant of the system (77) of linear equations in unknowns $c_a$ is non-zero. Hence the equation has only the trivial solution, i.e., cotangent values are linearly independent over $\mathbb{Q}$.

We turn to Maillet–Demyanenko determinant part of Theorem 1. The proof being almost verbatim to that of ([5], Theorem 1), we reproduce the proof and explain the steps.

$$\begin{array}{cc}\hfill D_k^0\left(0\right):& =det{\left({\tilde{B}}_k^0\left(\frac{\overline{a}{\overline{b}}^{-1}}{M}\right)\right)}_{\overline{a},\overline{b}\in T}\hfill \\ \hfill & =\prod _{\chi \in X^k}\sum _{\overline{a}\in T}\chi \left(\overline{a}\right){\tilde{B}}_k\left(\frac{\overline{a}}{M}\right)\hfill \\ \hfill & =\prod _{\chi \in X^k}\sum _{\overline{a}\in T}\sum _{\alpha \in \overline{a}H}\chi \left(\alpha \right){\tilde{B}}_k\left(\frac{R\left(\alpha \right)}{M}\right),\hfill \end{array}$$

where $R\left(\alpha \right)$ is the least non-negative residue mod M and $H=\left\{1\right\}$. Hence

$$\begin{array}{c}\hfill D_k^0\left(0\right)=\prod _{\chi \in X^k}\frac{1}{2}{\displaystyle \sum _{a=1}^{M-1}}\chi \left(a\right)B_k\left(\frac{a}{M}\right)=\prod _{\chi \in X^k}\frac{M^{1-k}}{2}{B}_{k,{\chi }^{*}}\prod _{\genfrac{}{}{0pt}{}{p|M}{p:prime}}\left(1-{\chi }^{*}\left(p\right)p^{k-1}\right),\end{array}$$

(82)

where $B_{k,{\chi }^{*}}$ is the generalized kth Bernoulli number defined by (72) and (72) is used in the last step.

Here we digress by appealing to the right-hand side of (2). For simplicity, we choose $k=1$. Then $X^1=X^o$ and we have by Lemma 4

$$\begin{array}{c}\hfill \frac{P(0,{\overline{\chi }}^{*})}{P(1,{\chi }^{*})}L(1,\chi )=P(0,{\overline{\chi }}^{*})L(1,{\chi }^{*})=i\frac{\pi \tau \left({\chi }^{*}\right)}{f}P(0,{\overline{\chi }}^{*}){\displaystyle \sum _{a=1}^{M-1}}\overline{{\chi }^{*}}\left(a\right){\overline{B}}_1\left(\frac{a}{M}\right).\end{array}$$

(83)

Then (82) becomes

$$\begin{array}{c}\hfill D_1^0\left(0\right)=-i\prod _{\chi \in X^o}\frac{f}{2\pi \tau \left({\chi }^{*}\right)}\frac{P(0,{\overline{\chi }}^{*})}{P(1,{\chi }^{*})}L(1,\chi )={\left(-\frac{1}{2\pi i}\right)}^{\frac{\phi \left(M\right)}{2}}{\delta }_1\prod _{\chi \in X^o}\tau \left({\overline{\chi }}^{*}\right)L(1,\chi ),\end{array}$$

(84)

where slightly correcting ([5] Theorem 3), we understand ${\delta }_k$ to mean

$${\delta }_k=\prod _{\chi \in X^o}\frac{P(1-k,{\overline{\chi }}^{*})}{P(k,{\chi }^{*})}.$$

(85)

Therefore, the essential ingredient is again the product ${\prod }_{\chi \in X^o}L(1,\chi )$. If all $\chi \in X^o$ are primitive, the situation will be much simpler. □

Remark 3. 

(i) Substituting (22) and using the (inverse) Eisenstein Formula (88), the right-hand side of (3) becomes

$$-\frac{\pi i}{M}{\displaystyle \sum _{a=1}^{M-1}}f\left(a\right){\displaystyle \sum _{k=1}^{M-1}}{\epsilon }_k(-a){\overline{B}}_1\left(\frac{k}{M}\right)=\frac{\pi }{2M}{\displaystyle \sum _{a=1}^{M-1}}f\left(a\right)cot\frac{\pi a}{M}$$

(86)

i.e., we just return to the left-hand side. It follows that the DFT formula alone does not lead to merging, and the Dedekind determinant formula is needed.

(ii) If we apply the functional equation (twice, cf. below)

$$\begin{array}{cc}\hfill & D(1,f^o)=i\frac{\pi }{\sqrt{M}}D(0,{\widehat{f}}^o)\hfill \\ \hfill & =i\frac{\pi }{\sqrt{M}}\left({\displaystyle \sum _{k=1}^M}\widehat{f}\left(k\right)\zeta \left(0,\frac{k}{M}\right)\right)=-i\frac{\pi }{\sqrt{M}}{\displaystyle \sum _{k=1}^{M-1}}\widehat{f}\left(k\right){\overline{B}}_1\left(\frac{k}{M}\right),\hfill \end{array}$$

(87)

then we come back to the same expression as is expected.

In Remark 3, (ii) we applied the (inverse) Eisenstein formula

$$\sqrt{M}{\widehat{\overline{B}}}_1\left(\frac{b}{M}\right)={\displaystyle \sum _{a=1}^{M-1}}{e}^{-2\pi i\frac{ab}{M}}{\overline{B}}_1\left(\frac{a}{M}\right)=-\frac{1}{2i}cot\frac{\pi b}{M},$$

(88)

(88) is the special case of (35) with $f\left(k\right)={\chi }_b\left(k\right)$, which in turn is contained in (36), i.e., the base change formula.

In Remark 3, we used the formulas

$$D(1,f^o)=i\frac{\pi }{\sqrt{M}}D(0,{\widehat{f}}^o).$$

(89)

and

$$\zeta (1-\varkappa ,\left\{x\right\})=-\frac{1}{\varkappa }{\overline{B}}_{\varkappa }\left(x\right)$$

(90)

valid for $\left\{x\right\}>0$.

Both follow from the functional equation in the generalized Euler identity form as in Theorem 6.

4. Appendix

Here we collect historical accounts on CPMD and information on zeta-allied functions.

4.1. Chowla’s Problem and Maillet–Demyanenko Determinants

Since [31] is very informative, including applications of [32] and rather thorough references up to 1986, we collect references that are not listed there. The following is the list of papers on Chowla’s problem to be added [1,2,30,33,34,35,36].

Ref. [37] is rather enlightening and clarifies that the rth derivative case of cotangent functions are rather trivial save for $r=0$ in which case [32] gives a more substantial non-vanishing result of $L(1,\chi )$.

We note that only Baker–Birch–Wirsing [32] resp. Fujiwara [34] attacked the original problem of Chowla, the former proved the Baker–Birch–Wirsing theorem, claiming that irreducibility of the Mth cyclotomic polynomial ${\Phi }_M$ implies the non-vanishing of $L(1,f)$. $\mathbb{Q}\left(f\right(1),\cdots ,f(M\left)\right)$ and the latter tried the theory of n-gons on the grounds of (95).

We quote the following theorems of Funakura, which are more detailed and general versions than those of Fujisaki [33], and clarify the situation.

Theorem 7 

(Ref. [20] Theorem 20-I). For a positive integer $M\ge 3$, the following are equivalent.

(i)

$$cot\frac{\pi a}{M},1\le a<\frac{M}{2},(a,M)=1$$

(91)

are linearly independent over $\mathbb{Q}$.

(ii) holds with cotangent replaced by tangent.

(iii) For every divisor d of M, $sin\frac{2\pi }{d}$ is expressed as a linear combination of cotangents (91) with rational coefficients.

(iv) holds with sin replaced by cosine.

(v)

$$L(1,\chi )\ne 0$$

(92)

for all odd characters, mod M.

Theorem 8 

(Ref. [20] Theorem 20-II). Suppose one of the conditions in Theorem 7 holds. Then for positive integer M, the following are equivalent.

(II-1) For every prime divisor $p|M$ the exponent $M\left(p\right)$ is even, where the exponent is defined by

$$M\left(p\right)=\left\{\begin{array}{cc}0\hfill & M=p^a\hfill \\ min\{m\in \mathbb{N}|p^m\equiv 1mod\frac{M}{p^a}\}\hfill & \mathit{otherwise}\hfill \end{array}\right.,$$

(93)

and where $p^a$ is the highest power of p dividing M.

(II-2) For χ odd, (118) holds.

(II-3) The degree $\frac{\phi \left(M\right)}{2}$ matrix

$$\mathcal{M}=(2\overline{a{b}^{-1}}-M)=\left(2M{\overline{B}}_1\left(\frac{a{b}^{-1}}{M}\right)\right)$$

(94)

is regular, where $a,b$ run over $1,\cdots ,\frac{\phi \left(M\right)}{2}$.

Assume M is odd. Then the above conditions are equivalent to either of the following two conditions.

(II-4) For χ odd,

$${\displaystyle \sum _{a=1}^{M-1}}{(-1)}^a\chi \left(a\right)\ne 0$$

(95)

holds.

(II-5) The degree $\frac{\phi \left(M\right)}{2}$ matrix

$$\mathcal{D}=({(-1)}^{\overline{a{b}^{-1}}})$$

(96)

is regular, where the conditions are the same as in(II-2).

Remark 4. 

(i) Theorems 7 and 8 are for the (inverse) Chowla problem. From the point of view of the original problem of Chowla the statement of Theorem 8 is not satisfactory, and Fujisaki’s version is more relevant: Suppose $M>1$ is odd and $(M,\phi (M\left)\right)=1$. Then statements (i)–(iv) of Theorem 7 are equivalent to statements (II-2) and (II-4) of Theorem 8. However, Maillet–Demyanenko determinants are restricted to the prime modulus.

(ii) Note that the determinant of the matrix in (II-3) resp. (II-5) is the Maillet resp. Demyanenko determinant.

We turn to the Maillet and Demyanenko determinants. Yamamura published an extensive list on MD up to 2008 in [38]. Now he has published the updated list, which may not be easily available, and we record those which are published after 2009 in the References. They are [38,39,40,41,42], in this order which is labelled. Papers [6,43,44,45,46] are to be added in the list.

Research on Maillet–Demyanenko determinants was started by Carlitz and Olson and reference to it is found in [43]. Hence [20,43] (a generalization of [33]) are the missing links that connect them.

The linkage can be perceived in a few papers. Milnor [13] is another essential source on the linkage describing the structure of the Kubert space of functions and referring to Chowlas’ problem. The work in [20] contains a prototype of Theorem 4 for the Laurent constant of a periodic Dirichlet series $L(s,f)$.

4.2. Cyclotomic Fields and Class Numbers

In this subsection, we briefly describe the underlying class-field-theoretic backbone that clarifies the setting of the Baker–Birch–Wirsing theorem as well as our main theorem, Theorem 1.

We quote from ([47] pp. 171–172). Prime Ideal Theorem in an Arithmetic Progression (PITAP).

Theorem 9 

(Theorem 1). For every congruence classification in an algebraic number field k, each congruence class contains infinitely many (absolute) prime ideals of degree 1, so that for the (Hecke) L-function corresponding to the classification, we have $L(1,{\chi }_i)\ne 0$, $\chi \ne {\chi }_1$

Thus $L(1,{\chi }_i)\ne 0$ is assured rather excessively by class field theory.

Fundamental references on class numbers of an Abelian field are [18,48]. For the relative class number (also referred to as the first factor), cf., e.g., [44]. For cyclotomic fields, we refer to [16,17].

The Mth cyclotomic polynomial ${\Phi }_M\left(X\right)\in \mathbb{Q}\left[X\right]$ is the degree $\phi \left(M\right)$ monic polynomial whose roots are primitive Mth roots ${\zeta }_M$, of unity. The field ${\mathbb{Q}}_M=\mathbb{Q}\left({\zeta }_M\right)$ is called the Mth cyclotomic field. We may assume that M is not of the form $2a$ with a odd. K is an Abelian extension of $\mathbb{Q}$ with its Galois group $\mathrm{Gal}({\mathbb{Q}}_M/\mathbb{Q})\cong {(\mathbb{Z}/M\mathbb{Z})}^{\times }$. Here the isomorphism is given by $\phi \left({\sigma }_s\right)=smodM$, each $\sigma \in \mathrm{Gal}(K/\mathbb{Q})$ being a raising-to-the-power s mod q map ${\sigma }_s$, $(s,q)=1$:

$${\zeta }^{\sigma }={\zeta }^s.$$

(97)

Using the Kronecker–Weber theorem, every Abelian extension K of $\mathbb{Q}$ is a subfield of ${\mathbb{Q}}_M$ and conversely. Regarding (4), ([49] p. 11, (3b)) reads

$$h_M^{-}=Qw{\displaystyle \prod _{\chi \in X^o}^{}}-\frac{1}{2f\left(\chi \right)}{\displaystyle \sum _{a=1}^{f\left(\chi \right)}}a\chi \left(a\right),$$

(98)

where the product is over all odd primitive characters $\chi$ with conductor $f\left(\chi \right)$ and $w=\left|W\right|$ and $Q=[E:W{E}^{+}]$, and where W is the group of roots of unity of K, E resp. $E^{+}$ is the unit group of K resp. $K^{+}$. This follows from class number formulas for K and its maximal real subfield $K^{+}$, which depends on the decomposition theorem of the Dedekind zeta function, cf., e.g., ([50] (345), p. 186).

Invoking (119), we have, as in ([46] Corollary 1), the relative class number formula

$$\prod _{\chi \in X^o}L(1,\chi )=\frac{{\left(2\pi \right)}^g{h}^{-}\left(K\right)}{Qw\sqrt{|\frac{d\left(K\right)}{d\left(K^{+}\right)}|}}.$$

(99)

4.3. Zeta Functions and Ramified Functional Equations

In this subsection, we assemble some basics of Hurwitz and Lerch zeta functions and the ramified functional equation (zeta-symmetry) between them, (103) and (104), which are the genesis of the corresponding theorem, Theorem 6 for periodic Dirichlet series. Standard references on special functions used here are [51,52,53] etc.

Let ${\ell }_s\left(x\right)$ be the (boundary) Lerch zeta function defined by

$${\ell }_s\left(x\right)={\displaystyle \sum _{n=1}^{\infty }}{e}^{2\pi ixn}{n}^{-s},\sigma >1\mathrm{or}\sigma >0,x\notin \mathbb{Z}$$

(100)

on which ([19] pp. 277–279) contains rich information. It has its counterpart, the Hurwitz zeta function

$$\zeta (s,x)={\displaystyle \sum _{n=0}^{\infty }}\frac{1}{{(n+x)}^s},\sigma >1.$$

(101)

This is continued meromorphically over the whole plane with a simple pole at $s=1$. Both of them reduce to

$$\zeta (s,1)={\ell }_s\left(1\right)=\zeta \left(s\right)$$

(102)

through which we introduce the Riemann zeta function. These are connected by the Hurwitz formula (i.e., the functional equation for the Hurwitz zeta function): for $\sigma >1,0<x\le 1$,

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

(103)

while its reciprocal is

$${\ell }_{1-s}\left(x\right)=\frac{\Gamma \left(s\right)}{{\left(2\pi \right)}^s}\left(e^{\frac{\pi is}{2}}\zeta (s,x)+e^{-\frac{\pi is}{2}}\zeta (s,1-x)\right),0<x<1.$$

(104)

The limiting case with $x\to 1$ of (103) amounts to

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

(105)

(103) and (104) are put in the form of generalized Euler identity [10], cf. (53).

The Euler digamma function

$$\psi \left(x\right)=\frac{{\Gamma }^{\prime }\left(x\right)}{\Gamma \left(x\right)}$$

(106)

plays a crucial role in the theory of periodic and L-functions, (35) and (50) in view of its Laurent expansion

$$\zeta (s,x)=\frac{1}{s-1}-\psi \left(x\right)+O(s-1),\mathrm{as}s\to 1,$$

(107)

whose special case reads

$$\zeta \left(s\right)=\frac{1}{s-1}+\gamma +O(s-1),\mathrm{as}s\to 1,$$

(108)

where $\gamma =-\psi \left(1\right)$ is the Euler constant.

For $1\le \varkappa \in \mathbb{Z}$ and $0<x<1$, we have

$${\ell }_{\varkappa }\left(x\right)=\frac{{\left(2\pi i\right)}^{\varkappa -1}}{\varkappa !}\left(A_{\varkappa }\left(x\right)-\pi i{B}_{\varkappa }\left(x\right)\right),$$

(109)

where the even part $A_{\varkappa }\left(x\right)$ is the Clausen function of order $\varkappa$

$$\begin{array}{cc}\hfill A_{\varkappa }\left(x\right)& =\frac{\varkappa !}{2{\left(2\pi i\right)}^{\varkappa -1}}{\displaystyle \underset{n=-\infty }{\overset{\infty }{\sum ^{\prime }}}}\frac{\mathrm{sgn}\left(n\right)e^{2\pi inx}}{n^{\varkappa }}\hfill \\ \hfill & =\frac{\varkappa !}{2{\left(2\pi i\right)}^{\varkappa -1}}\left({\ell }_{\varkappa }\left(x\right)+{(-1)}^{\varkappa -1}{\ell }_{\varkappa }(1-x)\right)\hfill \end{array}$$

(110)

while the odd part $B_{\varkappa }\left(x\right)$ is the periodic Bernoulli polynomial of order $\varkappa$

$${\overline{B}}_{\varkappa }\left(x\right)=-\frac{\varkappa !}{{\left(2\pi i\right)}^{\varkappa }}{\displaystyle \underset{n=-\infty }{\overset{\infty }{\sum ^{\prime }}}}\frac{e^{2\pi inx}}{n^{\varkappa }}=-\frac{\varkappa !}{{\left(2\pi i\right)}^{\varkappa }}\left({\ell }_{\varkappa }\left(x\right)-{(-1)}^{\varkappa -1}{\ell }_{\varkappa }(1-x)\right)$$

(111)

These are complementary to each other and satisfy the parity relation

$$A_{\varkappa }\left(x\right)={(-1)}^{\varkappa -1}{A}_{\varkappa }(1-x),{\overline{B}}_{\varkappa }\left(x\right)={(-1)}^{\varkappa }{\overline{B}}_{\varkappa }(1-x).$$

(112)

We assemble data on the Lerch zeta function ${\ell }_s\left(x\right)$, $s=0,1$ in (100) and the Euler digamma function $\psi \left(x\right)$, (106) for easy reference, cf. e.g., [19,22,51] etc., Under the Gaussian representation

$$\psi \left(x\right)=-\gamma -\frac{1}{x}+{\displaystyle \sum _{n=1}^{\infty }}\frac{x}{n(n+x)},$$

(113)

the following are all equivalent.

$$\begin{array}{cc}\hfill & {\displaystyle \sum _{n=1}^{\infty }}\frac{cos\left(2\pi nx\right)}{n}+i{\displaystyle \sum _{n=1}^{\infty }}\frac{sin\left(2\pi nx\right)}{n}={\ell }_1\left(x\right)\hfill \\ \hfill & =-log\left(1-e^{2\pi ix}\right)={\displaystyle \sum _{n=1}^{\infty }}\frac{e^{2\pi inx}}{n}=A_1\left(x\right)-\pi i{\overline{B}}_1\left(x\right),\hfill \end{array}$$

(114)

$0<x<1$, where

$$A_1\left(x\right):=-log2|sin\pi x|={\displaystyle \sum _{n=1}^{\infty }}\frac{cos\left(2\pi nx\right)}{n}$$

(115)

is its real part (even part), the first Clausen function (or the logsine function), and the imaginary part (odd part) is (111) with $\varkappa =1$, which reads

$$-2\pi i{\overline{B}}_1\left(x\right)={\ell }_1\left(x\right)-{\ell }_1(-x)=2i{\displaystyle \sum _{n=1}^{\infty }}\frac{sin\left(2\pi nx\right)}{n},0<x<1.$$

(116)

$$\begin{array}{cc}\hfill \psi \left(x\right)-\psi (1-x)& =2\pi i{\ell }_0\left(x\right)+\pi i=2\pi i\frac{e^{2\pi ix}}{1-e^{2\pi ix}}+\pi i\hfill \\ \hfill & =-\pi cot\pi x=-\frac{1}{x}-2x{\displaystyle \sum _{n=1}^{\infty }}\frac{1}{x^2-n^2},0<x<1.\hfill \end{array}$$

(117)

4.4. Weighted Character Sums

Weighted character sums have been considered extensively and [19] is the decisive work that has closed further research, cf. [54] for its elucidation and historical account.

Iwasawa ([29] p. 14) states that there is no elementary proof known of

$${\displaystyle \sum _{a=1}^{M-1}}a\chi \left(a\right)\ne 0$$

(118)

for an odd character $\chi$ to the modulus M.

Using analytic methods, we obtain the following theorem,

Theorem 10. 

Suppose χ is odd and primitive. Then we have

$${\displaystyle \sum _{a=1}^{M-1}}a\chi \left(a\right)=-\frac{\tau \left(\chi \right)}{\pi i}L(1,\overline{\chi }),$$

(119)

which is in conformity with ([17], Theorem 4.9, p. 37) and the first order Riesz sum is

$$\begin{array}{c}\hfill \sum _{a\le x}(x-a)\chi \left(a\right)=L(0,\chi )x+\frac{i\tau \left(\chi \right)}{2{\pi }^2\sqrt{M}}{\displaystyle \sum _{n=1}^{\infty }}\frac{\overline{\chi }\left(n\right)}{n^2}sin\left(2\pi xn\right).\end{array}$$

(120)

It is clear that (120) yields (119) in view of (71).