Multiple Dirichlet series associated to additive and Dirichlet characters

TL;DR

This paper investigates the analytic properties of multiple Dirichlet series linked to additive and Dirichlet characters, establishing meromorphic continuation and singularity sets for additive cases, and exploring connections for Dirichlet cases.

Contribution

It provides the first meromorphic continuation for multiple Dirichlet series associated to additive characters and analyzes their singularities, also relating these to Dirichlet characters.

Findings

Meromorphic continuation for additive character series

Exact singularity set for additive character series

Connection between additive and Dirichlet character series

Abstract

In this article we study analytic properties of the multiple Dirichlet series associated to additive and Dirichlet characters. For the multiple Dirichlet series associated to additive characters, the meromorphic continuation is established via obtaining translation formulas satisfied by these multiple Dirichlet series. While it seems difficult to obtain such a translation formula for the multiple Dirichlet series associated to Dirichlet characters, we rely on their intrinsic connection with the multiple Dirichlet series associated to additive characters in order to investigate their analytic characteristics. We are also able to determine the exact set of singularities of the multiple Dirichlet series associated to additive characters.

Equations

$$U_r:=\{(s_1,\dots ,s_r)\in {\mathbb{C}}^r:\Re (s_1+\cdots +s_i)>i\text{ for all }1\le i\le r\}.$$

$${\zeta }_r(s_1,\dots ,s_r):=\sum _{n_1>\cdots >n_r>0}{n}_1^{-s_1}\cdots n_r^{-s_r}.$$

$$s_1=1;s_1+\cdots +s_i=n\text{ for all }n\in {\mathbb{Z}}_{\le i}\text{ and }2\le i\le r.$$

$$\begin{array}{rl}& s_1=1;s_1+s_2=2,1,0,-2,-4,-6,\dots ;\\ & s_1+\cdots +s_i=n\text{ for all }n\in {\mathbb{Z}}_{\le i}\text{ and }3\le i\le r.\end{array}$$

$${\zeta }_{r-1}(s_1+s_2-1,s_3,\dots ,s_r)=\sum _{k\ge 0}(s_1-1{)}_k{\zeta }_r(s_1+k,s_2,\dots ,s_r),$$

$$(s{)}_k:=\frac{s\cdots (s+k)}{(k+1)!}.$$

$$L_r(s_1,\dots ,s_r;{\chi }_1,\dots ,{\chi }_r):=\sum _{n_1>\cdots >n_r>0}\frac{{\chi }_1(n_1)\cdots {\chi }_r(n_r)}{n_1^{s_1}\cdots n_r^{s_r}}.$$

$${\left(\frac{{\chi }_1(n_1)\cdots {\chi }_r(n_r)}{n_1^{s_1}\cdots n_r^{s_r}}\right)}_{n_1>\cdots >n_r>0}$$

$$s_1=1;s_1+\cdots +s_i=n\text{ for all }n\in {\mathbb{Z}}_{\le i}\text{ and }2\le i\le r.$$

$$\sum _{a=1}^q\frac{\chi (a)}{a^{(s-1)}}=\sum _{k\ge 0}(s-1{)}_k{q}^{k+1}\left(L(s+k;\chi )-\sum _{a=1}^q\frac{\chi (a)}{a^{(s+k)}}\right),$$

$$L(s;\chi ):=\sum _{n\ge 1}\frac{\chi (n)}{n^s},$$

$$\Phi (s;a):=\sum _{\begin{array}{c}n\ge 1\\ n\equiv a\mathrm{mod}N\end{array}}\frac{1}{n^s},$$

$$\Psi (s;b):=\sum _{n\ge 1}\frac{e^{2\pi \iota bn/N}}{n^s}.$$

$$L(s;\chi )=\sum _{1\le a\le N}\chi (a)\Phi (s;a),$$

$$\Psi (s;b)=\sum _{1\le a\le N}{e}^{2\pi \iota ab/N}\Phi (s;a).$$

$$\Phi (s;a)=\frac{1}{N}\sum _{1\le b\le N}{e}^{-2\pi \iota ab/N}\Psi (s;b).$$

$$L(s;\chi )=\frac{1}{N}\sum _{1\le a\le N}\chi (a)\sum _{1\le b\le N}{e}^{-2\pi \iota ab/N}\Psi (s;b).$$

$$L_r(s_1,\dots ,s_r;{\chi }_1,\dots ,{\chi }_r)=\sum _{\begin{array}{c}1\le a_i\le N_i\\ \text{for all }1\le i\le r\end{array}}{\chi }_1(a_1)\cdots {\chi }_r(a_r){\Phi }_r(s_1,\dots ,s_r;a_1,\dots ,a_r),$$

$${\Phi }_r(s_1,\dots ,s_r;a_1,\dots ,a_r):=\sum _{\begin{array}{c}{n}_1>\cdots >n_r>0\\ n_i\equiv a_i\mathrm{mod}{N}_i\\ \text{for all }1\le i\le r\end{array}}{n}_1^{-s_1}\cdots n_r^{-s_r}.$$

$${\Psi }_r(s_1,\dots ,s_r;b_1,\dots ,b_r):=\sum _{n_1>\cdots >n_r>0}\frac{e^{2\pi \iota \left(\frac{b_1{n}_1}{N_1}+\cdots +\frac{b_r{n}_r}{N_r}\right)}}{n_1^{s_1}\cdots n_r^{s_r}}.$$

$${\Psi }_r(s_1,\dots ,s_r;b_1,\dots ,b_r)=\sum _{\begin{array}{c}1\le a_i\le N_i\\ \text{for all }1\le i\le r\end{array}}{e}^{2\pi \iota \left(\frac{a_1{b}_1}{N_1}+\cdots +\frac{a_r{b}_r}{N_r}\right)}{\Phi }_r(s_1,\dots ,s_r;a_1,\dots ,a_r).$$

$$\begin{array}{rl}& {\Phi }_r(s_1,\dots ,s_r;a_1,\dots ,a_r)\\ & =\frac{1}{N_1\cdots N_r}\sum _{\begin{array}{c}1\le b_i\le N_i\\ \text{for all }1\le i\le r\end{array}}{e}^{-2\pi \iota \left(\frac{a_1{b}_1}{N_1}+\cdots +\frac{a_r{b}_r}{N_r}\right)}{\Psi }_r(s_1,\dots ,s_r;b_1,\dots ,b_r).\end{array}$$

$$\begin{array}{rl}& L_r(s_1,\dots ,s_r;{\chi }_1,\dots ,{\chi }_r)\\ & =\frac{1}{N_1\cdots N_r}\sum _{\begin{array}{c}1\le a_i\le N_i\\ \text{for all }1\le i\le r\end{array}}{\chi }_1(a_1)\cdots {\chi }_r(a_r)\sum _{\begin{array}{c}1\le b_i\le N_i\\ \text{for all }1\le i\le r\end{array}}{e}^{-2\pi \iota \left(\frac{a_1{b}_1}{N_1}+\cdots +\frac{a_r{b}_r}{N_r}\right)}{\Psi }_r(s_1,\dots ,s_r;b_1,\dots ,b_r).\end{array}$$

$$L_r(f_1,\dots ,f_r;s_1,\dots ,s_r):=\sum _{n_1>\cdots >n_r>0}\frac{f_1(n_1)\cdots f_r(n_r)}{n_1^{s_1}\cdots n_r^{s_r}}=\sum _{n_1>\cdots >n_r>0}\frac{f_1(1{)}^{n_1}\cdots f_r(1{)}^{n_r}}{n_1^{s_1}\cdots n_r^{s_r}}.$$

$$|g_i(1)|\le 1\text{for all}1\le i\le r.$$

$$\left|f_1(1{)}^{n_1}\cdots f_r(1{)}^{n_r}\right|\le \left|g_1(1{)}^{n_1-n_2}\cdots g_{r-1}(1{)}^{n_{r-1}-n_r}{g}_r(1{)}^{n_r}\right|.$$

$$\sum _{n_1>\cdots >n_r>0}\frac{f_1(n_1)\cdots f_r(n_r)}{n_1^{s_1}\cdots n_r^{s_r}}$$

$$\sum _{n_1>\cdots >n_r>0}\frac{|f_1(n_1)\cdots f_r(n_r)|}{n_1^{{\sigma }_1}\cdots n_r^{{\sigma }_r}}$$

$$\sum _{n>r-i}\frac{|f_1(n+i-1)f_2(n+i-2)\cdots f_i(n)f_{i+1}(r-i)\cdots f_r(1)|}{(n+i-1{)}^{{\sigma }_1}(n+i-2{)}^{{\sigma }_2}\cdots n^{{\sigma }_i}(r-i{)}^{{\sigma }_{i+1}}\cdots 1^{{\sigma }_r}}$$

$$|g_1(1)\cdots g_{i-1}(1)g_i(1{)}^n{f}_{i+1}(r-i)\cdots f_r(1)|$$

$$(n+i-1{)}^{{\sigma }_1+\cdots +{\sigma }_i}(r-i{)}^{{\sigma }_{i+1}}\cdots 1^{{\sigma }_r}.$$

Full text

1. Introduction

Perhaps the most classical and the most studied multiple Dirichlet series are the so-called multiple zeta functions. For an integer $r\geq 1$, consider the open subset $U_{r}$ of ${\mathbb{C}}^{r}$:

[TABLE]

The multiple zeta function of depth $r$, denoted by $\zeta_{r}(s_{1},\ldots,s_{r})$, is a function on $U_{r}$ defined by

[TABLE]

The above series converges normally on any compact subset of $U_{r}$ (see [4] for a proof) and hence defines an analytic function on $U_{r}$. This is a multi-variable generalisation of the classical Riemann zeta function. The origin of these functions can be traced back to Euler, who studied the case $r=2$.

The analytic theory of the multiple zeta functions has been in the focus of research in recent years. As a function of several variables, the meromorphic continuation of the multiple zeta function of depth $r$ was first established by J. Zhao [7] in 1999. He used the theory of generalised functions.

Theorem 1** (Zhao).**

The multiple zeta function of depth $r$ can be extended as a meromorphic function to ${\mathbb{C}}^{r}$ with possible simple poles at the hyperplanes given by the equations

[TABLE]

Here ${\mathbb{Z}}_{\leq i}$ denotes set of all integers less than or equal to $i$.

In about the same time, S. Akiyama, S. Egami and Y. Tanigawa [1] gave a simpler proof of the above fact using the classical Euler-Maclaurin summation formula. They could also identify the exact set of polar hyperplanes. The vanishing of the odd Bernoulli numbers plays a central role in this context.

Theorem 2** (Akiyama-Egami-Tanigawa).**

The multiple zeta function of depth $r$ is holomorphic in the open set obtained by removing the following hyperplanes from ${\mathbb{C}}^{r}$ and it has simple poles at the hyperplanes given by the equations

[TABLE]

Thereafter, the problem of the meromorphic continuation of the multiple zeta functions drew attention of several mathematicians. In this process, a variety of methods evolved to address this problem. Perhaps the simplest approach to this problem has been exhibited in [4], where the meromorphic continuation of the multiple zeta functions is obtained by means of a simple translation formula and induction on the depth $r$.

Theorem 3** (Mehta-Saha-Viswanadham).**

For each integer $r\geq 2$, the multiple zeta function of depth $r$ extends to a meromorphic function on $\mathbb{C}^{r}$ satisfying the translation formula

[TABLE]

where the series of meromorphic functions on the right hand side converges normally on all compact subsets of ${\mathbb{C}}^{r}$ and for any $k\geq 0$ and $s\in{\mathbb{C}}$,

[TABLE]

Further in [4], the method of matrix formulation of translation formulas was introduced to write down the residues along the possible polar hyperplanes in a computable form and recover the above mentioned theorem of Akiyama, Egami and Tanigawa.

Soon after the works of Zhao [7], Akiyama, Egami and Tanigawa [1], several generalisations of the multiple zeta functions were introduced and their analytic properties were discussed. One such important generalisation is the multiple Dirichlet series associated to Dirichlet characters. It was first introduced by Akiyama and Ishikawa [2] in 2002. Multiple Dirichlet series associated to Dirichlet characters are now known as multiple Dirichlet $L$-functions as these are several variable generalisations of the classical Dirichlet $L$-functions.

Definition 1**.**

Let $r\geq 1$ be an integer and $\chi_{1},\ldots,\chi_{r}$ be Dirichlet characters of arbitrary modulus. The multiple Dirichlet $L$-function of depth $r$ associated to the Dirichlet characters $\chi_{1},\ldots,\chi_{r}$ is denoted by $L_{r}(s_{1},\ldots,s_{r};~{}\chi_{1},\ldots,\chi_{r})$ and defined by the following normally convergent series in $U_{r}$:

[TABLE]

Akiyama and Ishikawa introduced the notion of multiple Dirichlet $L$-functions for characters with same modulus, but their definition also makes sense for Dirichlet characters of arbitrary modulus. The normal convergence of the above series follows from the normal convergence of the multiple zeta function of depth $r$ as an immediate consequence and we record it here in the following proposition. Throughout this article, whenever we consider a set of characters, they are of arbitrary modulus unless otherwise stated.

Proposition 1**.**

Let $r\geq 1$ be an integer and $\chi_{1},\ldots,\chi_{r}$ be Dirichlet characters. Then the family of functions

[TABLE]

is normally summable on compact subsets of $U_{r}$.

So it follows that $(s_{1},\ldots,s_{r})\mapsto L_{r}(s_{1},\ldots,s_{r};~{}\chi_{1},\ldots,\chi_{r})$ is a holomorphic function on $U_{r}$. Akiyama and Ishikawa have also discussed the question of meromorphic continuation of the multiple Dirichlet $L$-functions. When $r=1$, classically the meromorphic continuation is achieved by writing the function in terms of the Hurwitz zeta function. For $r>1$, they pursued this idea under the assumption that all the characters are of same modulus. In this process some variants of the multiple Hurwitz zeta functions came up and they proved the following theorem.

Theorem 4** (Akiyama-Ishikawa).**

Let $\chi_{1},\ldots,\chi_{r}$ be Dirichlet characters of same modulus. Then the multiple Dirichlet $L$-function $L_{r}(s_{1},\ldots,s_{r};~{}\chi_{1},\ldots,\chi_{r})$ of depth $r$ can be extended as a meromorphic function to ${\mathbb{C}}^{r}$ with possible simple poles at the hyperplanes given by the equations

[TABLE]

At present, obtaining a complete description of the exact set of singularities of the multiple Dirichlet $L$-functions seems to be a difficult problem. For $r=2$ and specific choices of the characters $\chi_{1}$ and $\chi_{2}$, Akiyama and Ishikawa were able to determine the exact set of polar hyperplanes. To address this difficult question, one can aim to obtain a translation formula satisfied by the multiple Dirichlet $L$-functions. For $r=1$, such a translation formula has been established in [5].

Theorem 5** (Saha).**

Let $\chi$ be a Dirichlet character of modulus $q$. Then for $\Re(s)>1$, the associated Dirichlet $L$-function $L(s;\chi):=\sum_{n\geq 1}\frac{\chi(n)}{n^{s}}$ satisfies the following translation formula:

[TABLE]

where the infinite series on the right hand side converges normally on every compact subset of $\Re(s)>1$.

As mentioned earlier, the meromorphic continuation of $L(s;\chi)$ is generally derived by writing $L(s;\chi)$ in terms of the Hurwitz zeta function and then using the meromorphic continuation of the Hurwitz zeta function. But one can also derive this using Theorem 5. However, obtaining such a translation formula for multiple Dirichlet $L$-functions seems harder and so is the analogue of Theorem 5 for the multiple Dirichlet $L$-functions.

Now one can consider additive characters, i.e. group homomorphisms $f:{\mathbb{Z}}\to{\mathbb{C}}^{*}$, in place of Dirichlet characters. It is interesting to note that Dirichlet characters are linked to additive characters and so is the Dirichlet $L$-functions to the Dirichlet series associated to additive characters. Thus studying multiple Dirichlet series associated to additive characters are very much relevant to the study of multiple Dirichlet L-functions. In the following section we introduce the notion of multiple Dirichlet series associated to additive characters, which we call as multiple additive $L$-functions and explore their relation with multiple Dirichlet $L$-functions.

In this article, we obtain the meromorphic continuation of the multiple additive $L$-functions (see Theorem 8), by means of translation formulas satisfied by these multiple additive $L$-functions. Following the method of matrix formulation of translation formulas of [4], we first obtain an expression for residues along possible polar hyperplanes of these multiple additive $L$-functions (see Theorem 10). We then determine the exact set of singularities of these multiple additive $L$-functions (see Theorem 12). For this we require an important theorem of Frobenius about the zeros of the Eulerian polynomials (see §6.3). Further, using the relation between the multiple additive $L$-functions and the multiple Dirichlet $L$-functions, we extend the above theorem of Akiyama and Ishikawa for characters of arbitrary modulus (see Theorem 13).

2. Multiple additive $L$-functions

2.1. Relation with multiple Dirichlet $L$-functions

Let $a,b,N$ be natural numbers with $1\leq a,b\leq N$ and $\chi$ denote a Dirichlet character of modulus $N$. For $\Re(s)>1$, we consider the following Dirichlet series:

[TABLE]

Then we have the following relations among these Dirichlet series:

[TABLE]

[TABLE]

Further using (4), one can deduce that

[TABLE]

Hence

[TABLE]

In fact (6) can be generalised for multiple Dirichlet $L$-functions. Let $r\geq 1$ be a natural number and for each $1\leq i\leq r$, let $a_{i},b_{i},N_{i}$ be natural numbers with $1\leq a_{i},b_{i}\leq~{}N_{i}$. Also for each $i$, let $\chi_{i}$ be a Dirichlet character of modulus $N_{i}$. Then the depth-$r$ multiple Dirichlet $L$-function associated to $\chi_{1},\ldots,\chi_{r}$ can be written as follows:

[TABLE]

where $\Phi_{r}(s_{1},\ldots,s_{r};a_{1},\ldots,a_{r})$ is defined as the following multiple Dirichlet series in $U_{r}$:

[TABLE]

Next we consider the following multiple Dirichlet series in $U_{r}$:

[TABLE]

Now we can write down a several variable generalisation of (4):

[TABLE]

Further using (8), we derive

[TABLE]

Therefore, using (7) and (9) we obtain that

[TABLE]

The above formula (10) plays an important role in the investigation of the analytic properties of the multiple Dirichlet $L$-functions.

2.2. Multiple additive $L$-functions and their translation formulas

We now formally introduce the notion of multiple additive $L$-functions. Besides the above exhibited relation (10) with the multiple Dirichlet $L$-functions, the multiple additive $L$-functions are also of independent interest. Special values of these functions, such as the so-called cyclotomic multiple zeta values, the coloured multiple zeta values have also been studied in arithmetic context. These functions can also be thought of as cousins of the multiple polylogarithms.

Definition 2**.**

For a natural number $r\geq 1$ and additive characters $f_{1},\ldots,f_{r}$, the multiple additive $L$-function associated to $f_{1},\ldots,f_{r}$ is denoted by $L_{r}(f_{1},\ldots,f_{r};~{}s_{1},\ldots,s_{r})$ and defined by the following series:

[TABLE]

A necessary and sufficient condition for the absolute convergence of the above series in $U_{r}$ is given in terms of the partial products $g_{i}:=\prod_{1\leq j\leq i}f_{j}$ for all $1\leq i\leq r$. The condition is that

[TABLE]

The sufficiency of this condition is easily established by noting that for any arbitrary $r$ integers $n_{1}>\cdots>n_{r}>0$, one has

[TABLE]

It can also be shown that if $|g_{i}(1)|>1$ for some $i$, the above series does not converge absolutely in $U_{r}$, in fact, anywhere in ${\mathbb{C}}^{r}$. To see this, let $i$ be one of the index such that $|g_{i}(1)|>1$. Now if possible let for a complex $r$-tuple $(s_{1},\ldots,s_{r})$, the series

[TABLE]

converges absolutely i.e. the series

[TABLE]

of non-negative real numbers converges, where $\sigma_{i}$ denotes the real part of $s_{i}$. Then the following smaller series

[TABLE]

is convergent. Note that the numerator of the summand in the above series is nothing but

[TABLE]

and the denominator is smaller than

[TABLE]

As the series

[TABLE]

does not converge for any choice of complex $r$-tuple $(s_{1},\ldots,s_{r})$ when $|g_{i}(1)|>1$, we get a contradiction.

If we write $f_{i}(1)=e^{2\pi\iota\lambda_{i}}$ for some $\lambda_{i}\in{\mathbb{C}}$, the condition $|g_{i}(1)|\leq 1$ for $1\leq i\leq r$ can be rewritten as

[TABLE]

With the necessary and sufficient condition for absolute convergence of the multiple additive $L$-functions in place, we derive the following result as an immediate consequence of the normal convergence of the multiple zeta function of depth $r$.

Proposition 2**.**

Let $r\geq 1$ be an integer and $f_{1},\ldots,f_{r}$ be additive characters such that the partial products $g_{i}:=\prod_{1\leq j\leq i}f_{j}$ satisfy the condition $|g_{i}(1)|\leq 1$ for all $1\leq i\leq r$. Then the family of functions

[TABLE]

is normally summable on compact subsets of $U_{r}$.

For sake of completeness, we recall the definition of normal convergence.

Definition 3**.**

Let $X$ be a set and $(f_{i})_{i\in I}$ be a family of complex valued functions defined on $X$. We say that the family of functions $(f_{i})_{i\in I}$ is normally summable on $X$ or the series $\sum_{i\in I}f_{i}$ converges normally on $X$ if

[TABLE]

and the family of real numbers $(\|f_{i}\|_{X})_{i\in I}$ is summable.

Definition 4**.**

Let $X$ be an open subset of ${\mathbb{C}}^{r}$ and $(f_{i})_{i\in I}$ be a family of meromorphic functions on $X$. We say that $(f_{i})_{i\in I}$ is normally summable or $\sum_{i\in I}f_{i}$ is normally convergent on all compact subsets of $X$ if for any compact subset $K$ of $X$, there exists a finite set $J\subset I$ such that each $f_{i}$ for $i\in I\setminus J$ is holomorphic in an open neighbourhood of $K$ and the family $(f_{i}|K)_{i\in{I\setminus J}}$ is normally summable on $K$. In this case, $\sum_{i\in I}f_{i}$ is a well defined meromorphic function on $X$.

Proposition 2 implies that for a natural number $r\geq 1$ and additive characters $f_{1},\ldots,f_{r}$ such that the partial products $g_{i}:=\prod_{1\leq j\leq i}f_{j}$ satisfy the condition $|g_{i}(1)|\leq 1$ for all $1\leq i\leq r$, the multiple additive $L$-function $L_{r}(f_{1},\ldots,f_{r};~{}s_{1},\ldots,s_{r})$ defines an analytic function in $U_{r}$. We now address the question of meromorphic continuation of such multiple additive $L$-functions. This is done via obtaining translation formulas analogous to (1). The multiple additive $L$-function $L_{r}(f_{1},\ldots,f_{r};~{}s_{1},\ldots,s_{r})$ satisfies the following translation formulas depending on the condition that whether $f_{1}(1)=1$ or not.

Theorem 6**.**

For any integer $r\geq 2$ and additive characters $f_{1},\ldots,f_{r}$ such that the partial products $g_{i}:=\prod_{1\leq j\leq i}f_{j}$ satisfy the condition $|g_{i}(1)|\leq 1$ for all $1\leq i\leq r$ with $f_{1}(1)=1$, the associated multiple additive $L$-function $L_{r}(f_{1},\ldots,f_{r};~{}s_{1},\ldots,s_{r})$ satisfies the following translation formula in $U_{r}$:

[TABLE]

where the series on the right hand side converges normally on any compact subset of $U_{r}$ .

Note that the translation formula (11) is nothing but the translation formula (1), written for $L_{r}(f_{1},\ldots,f_{r};~{}s_{1},\ldots,s_{r})$ in place of $\zeta_{r}(s_{1},\ldots,s_{r})$. If $f_{1}(1)\neq 1$, then we have the following theorem.

Theorem 7**.**

Let $r\geq 2$ be an integer and $f_{1},\ldots,f_{r}$ be additive characters such that the partial products $g_{i}:=\prod_{1\leq j\leq i}f_{j}$ satisfy the condition $|g_{i}(1)|\leq 1$ for all $1\leq i\leq r$ with $f_{1}(1)\neq 1$. Then the associated multiple additive $L$-function $L_{r}(f_{1},\ldots,f_{r};~{}s_{1},\ldots,s_{r})$ satisfies the following translation formula in $U_{r}$:

[TABLE]

where the series on the right side converges normally on any compact subset of $U_{r}$ .

3. Proof of the translation formulas

To prove Theorem 6, we need the identity which is valid for any integer $n\geq 2$ and any complex number $s$:

[TABLE]

Whereas to prove Theorem 7, we need the following version of (13), obtained by replacing $s$ with $s+1$ in (13):

[TABLE]

The identity (13) is obtained by writing the left hand side as $n^{1-s}\left((1-\frac{1}{n})^{1-s}-1\right)$ and expanding $(1-\frac{1}{n})^{1-s}$ as a Taylor series in $\frac{1}{n}$. Besides, we need the following proposition.

Proposition 3**.**

Let $r\geq 2$ be an integer and $f_{1},\ldots,f_{r}$ be additive characters such that the partial products $g_{i}:=\prod_{1\leq j\leq i}f_{j}$ satisfy the condition $|g_{i}(1)|\leq 1$ for all $1\leq i\leq r$. Then the family of functions

[TABLE]

is normally summable on compact subsets of $U_{r}$.

Proof.

From the hypothesis we have

[TABLE]

Now let $K$ be a compact subset of $U_{r}$ and set

[TABLE]

Since $r\geq 2$, for any strictly decreasing sequence $n_{1},\ldots,n_{r}$ of $r$ positive integers, we have $n_{1}\geq 2$. Hence for $k\geq 0$, we get

[TABLE]

Thus the proof follows from the normal convergence of the multiple zeta function of depth $r$ and the fact that for any real number $a$, the series

[TABLE]

is convergent. ∎

We are now ready to prove Theorem 6 and Theorem 7.

3.1. Proof of Theorem 6

We replace $n,s$ by $n_{1},s_{1}$ in (13) and multiply $\frac{f_{1}(n_{1})\cdots f_{r}(n_{r})}{n_{2}^{s_{2}}\cdots n_{r}^{s_{r}}}$ to both sides of (13) and obtain that

[TABLE]

Now we sum for $n_{1}>\cdots>n_{r}>0$. Since $f_{1}(1)=1$, using Proposition 3, we get

[TABLE]

This together with Proposition 3 completes the proof. ∎

3.2. Proof of Theorem 7

For this we replace $n,s$ by $n_{1},s_{1}$ in (14) and multiply $\frac{f_{1}(n_{1})\cdots f_{r}(n_{r})}{n_{2}^{s_{2}}\cdots n_{r}^{s_{r}}}$ to both sides of (14) and obtain that

[TABLE]

As before, we sum for $n_{1}>\cdots>n_{r}>0$ and use Proposition 3 with $s_{1}$ replaced by $s_{1}+1$. Since $f_{1}(1)\neq 1$, we obtain

[TABLE]

This together with Proposition 3 completes the proof. ∎

4. Meromorphic continuation

In this section, we establish the meromorphic continuation of the multiple additive $L$-functions using the translation formulas (11) and (12).

Theorem 8**.**

Let $r\geq 2$ be an integer and $f_{1},\ldots,f_{r}$ be additive characters such that the partial products $g_{i}:=\prod_{1\leq j\leq i}f_{j}$ satisfy the condition $|g_{i}(1)|\leq 1$ for all $1\leq i\leq r$. Then the associated multiple additive $L$-function $L_{r}(f_{1},\ldots,f_{r};~{}s_{1},\ldots,s_{r})$ extends to a meromorphic function on ${\mathbb{C}}^{r}$ satisfying the following translation formulas on ${\mathbb{C}}^{r}$:

[TABLE]

and

[TABLE]

The series of meromorphic functions on the right hand sides of (15) and (16) converge normally on every compact subset of ${\mathbb{C}}^{r}$.

Remark 1**.**

When $r=1$ and $f_{1}(1)=1$, the associated additive $L$-function is nothing but the Riemann zeta function, for which we know that it can be meromorphically continued to ${\mathbb{C}}$ with only a simple pole at $s=1$ with residue $1$. When $r=1$ and $f_{1}(1)\neq 1$, it is also known that the associated additive $L$-function has an analytic continuation to the entire complex plane (for a proof see [5]).

We prove Theorem 8 by induction on depth $r$. Assuming induction hypothesis for multiple additive $L$-functions of depth $(r-1)$, we first extend the multiple additive $L$-function $L_{r}(f_{1},\ldots,f_{r};~{}s_{1},\ldots,s_{r})$ as a meromorphic function to $U_{r}(m)$ for each $m\geq 0$, where

[TABLE]

Since open sets of the form $U_{r}(m)$ form an open cover of ${\mathbb{C}}^{r}$, we get the coveted meromorphic continuation to ${\mathbb{C}}^{r}$. In this case we need the following variant of Proposition 3.

Proposition 4**.**

Let $r\geq 2$ be an integer and $f_{1},\ldots,f_{r}$ be additive characters such that the partial products $g_{i}:=\prod_{1\leq j\leq i}f_{j}$ satisfy the condition $|g_{i}(1)|\leq 1$ for all $1\leq i\leq r$. Then the family of functions

[TABLE]

is normally summable on compact subsets of $U_{r}(m)$.

Proof.

Let $K$ be a compact subset of $U_{r}(m)$ and $S:=\sup_{(s_{1},\ldots,s_{r})\in K}|s_{1}|$. Then for $k\geq m-1$,

[TABLE]

Now as $(s_{1},\ldots,s_{r})$ varies over $U_{r}(m)$, $(s_{1}+m,\ldots,s_{r})$ varies over $U_{r}$. Then the proof follows from the normal convergence of the multiple zeta function of depth $r$ as the series $\sum_{k\geq m-1}\frac{(S)_{k}}{2^{k-m+1}}$ converges. ∎

4.1. Proof of Theorem 8

As mentioned before, we prove this theorem by induction on depth $r$. For $r=2$, the left hand sides of (15) and (16) have a meromorphic continuation to ${\mathbb{C}}^{2}$ by Remark 1. If $r\geq 3$, then the left hand side of (15) and the first term in the left hand side of (16) have a meromorphic continuation to ${\mathbb{C}}^{r}$ by the induction hypothesis.

We now establish the meromorphic continuation of the multiple additive $L$-function $L_{r}(f_{1},\ldots,f_{r};~{}s_{1},\ldots,s_{r})$ separately for each of the cases $f_{1}(1)=1$ and $f_{1}(1)\neq 1$.

First we consider the case $f_{1}(1)=1$. As we have shown that in this case the multiple additive $L$-function $L_{r}(f_{1},\ldots,f_{r};~{}s_{1},\ldots,s_{r})$ satisfies the translation formula (11) in $U_{r}$. Note that the translation formula (11) is exactly the same as (1) which is satisfied by the multiple zeta functions of depth $r$. Hence in this case the meromorphic continuation follows exactly as in the case of the multiple zeta function and therefore we omit the proof.

Next we consider the case $f_{1}(1)\not=1$. Now from induction hypothesis we know that

[TABLE]

has a meromorphic continuation to ${\mathbb{C}}^{r}$ and by Proposition 4 (for $m=1$), we have that

[TABLE]

is a holomorphic function on $U_{r}(1)$. Hence we can extend $L_{r}(f_{1},\ldots,f_{r};~{}s_{1},\ldots,s_{r})$ to $U_{r}(1)$ as a meromorphic function satisfying (12).

Now we assume that for an integer $m\geq 2$, $L_{r}(f_{1},\ldots,f_{r};~{}s_{1},\ldots,s_{r})$ has been extended meromorphically to $U_{r}(m-1)$ satisfying (12). Again by Proposition 4, the series

[TABLE]

defines a holomorphic function on $U_{r}(m)$. As $L_{r}(f_{1},\ldots,f_{r};~{}s_{1},\ldots,s_{r})$ has been extended meromorphically to $U_{r}(m-1)$, we can extend $L_{r}(f_{1},\ldots,f_{r};~{}s_{1}+k+1,\ldots,s_{r})$ meromorphically to $U_{r}(m)$ for all $0\leq k\leq m-2$.

Thus we can now extend $L_{r}(f_{1},\ldots,f_{r};~{}s_{1},\ldots,s_{r})$ meromorphically to $U_{r}(m)$ by means of (12). This completes the proof as $\{U_{r}(m):m\geq 1\}$ is an open cover of ${\mathbb{C}}^{r}$. ∎

5. Matrix formulation of the translation formulas

The method of matrix formulation of translation formulas was introduced in [4] in order to write down the translation formula (1) in an expression involving infinite matrices. We adapt this method to obtain similar matrix formulation of the translation formulas (15) and (16). More details about the method of matrix formulation of translation formulas can be found in [4].

5.1. Matrix formulation of (15)

We recall that the translation formula (15) is nothing but the translation formula (1), written for $L_{r}(f_{1},\ldots,f_{r};~{}s_{1},\ldots,s_{r})$ in place of $\zeta_{r}(s_{1},\ldots,s_{r})$. Thus the matrix formulation for (15) will be identical to the one for (1). For sake of completeness we discuss this in brief.

The translation formula (15) together with the other relations obtained by applying successively the change of variables $s_{1}\mapsto s_{1}+n$ to it for each $n\geq 0$ is equivalent to the single relation

[TABLE]

where ${\bf V}_{r}(f_{1},\ldots,f_{r};\ s_{1},\ldots,s_{r})$ denotes the infinite column vector

[TABLE]

and for an indeterminate $t$, ${\bf A_{1}}(t)$ is defined by

[TABLE]

The matrix ${\bf A_{1}}(t)$ is invertible in $\mathbf{T}({\mathbb{Q}}(t))$ and its inverse matrix ${\bf B_{1}}(t)$ is given by

[TABLE]

Here $\mathbf{T}({\mathbb{Q}}(t))$ denotes the ring of upper triangular matrices of type $\mathbb{N}\times\mathbb{N}$ with coefficients in ${\mathbb{Q}}(t)$.

This suggests that one may attempt to express the column vector ${\bf V}_{r}(f_{1},\ldots,f_{r};\ s_{1},\ldots,s_{r})$ in terms of ${\bf V}_{r-1}(f_{2},f_{3},\ldots,f_{r};\ s_{1}+s_{2}-1,s_{3},\ldots,s_{r})$ by multiplying both sides of (17) by ${\bf B_{1}}(s_{1}-1)$. However this is not allowed. The reason is that the entries of the formal product of the matrix ${\bf B_{1}}(s_{1}-1)$ with the column vector ${\bf V}_{r-1}(s_{1}+s_{2}-1,s_{3},\ldots,s_{r})$ are not convergent series.

To get around this difficulty we choose an integer $q\geq 1$ and define

[TABLE]

This allows us to write the previous matrices as block matrices, for example

[TABLE]

With these notations, we rewrite (17) as follows:

[TABLE]

Now inverting ${\bf A_{1}}^{II}(s_{1}-1)$, we can reformulate (17) to write

[TABLE]

where

[TABLE]

All the series of meromorphic functions involved in the products of matrices in formulas (21) and (22) converge normally on all compact subsets of ${\mathbb{C}}^{r}$. Moreover, all the entries of the matrices on the right hand side of (22) are holomorphic in the open set $U_{r}(q)$, the translate of $U_{r}$ by $(-q,0,\ldots,0)$. Therefore the entries of ${\bf W}^{I}(f_{1},\ldots,f_{r};\ s_{1},\ldots,s_{r})$ are also holomorphic in $U_{r}(q)$.

If we write $\xi_{q}(f_{1},\ldots,f_{r};\ s_{1},\ldots,s_{r})$ to be the first entry of ${\bf W}^{I}(f_{1},\ldots,f_{r};\ s_{1},\ldots,s_{r})$, we then get from (21) that

[TABLE]

where $\xi_{q}(f_{1},\ldots,f_{r};\ s_{1},\ldots,s_{r})$ is holomorphic in the open set $U_{r}(q)$.

5.2. Matrix formulation of (16)

On the other hand, the translation formula (16) and the other relations obtained by applying successively the change of variable $s_{1}\mapsto s_{1}+n$ to it for each $n\geq 0$, is equivalent to the relation

[TABLE]

where for an indeterminate $t$ and a non-trivial group homomorphism $f:{\mathbb{Z}}\to{\mathbb{C}}^{*}$, the matrix ${\bf A_{2}}(f;t)$ is defined as follows:

[TABLE]

As $f(1)\neq 1$, the matrix ${\bf A_{2}}(f;t)$ is invertible in ${\bf T}({\mathbb{C}}(t))$, the ring of upper triangular matrices of type ${\mathbb{N}}\times{\mathbb{N}}$ with coefficients in ${\mathbb{C}}(t)$. To find the inverse of ${\bf A_{2}}(f;t)$ explicitly, we notice that

[TABLE]

where ${\bf I}_{{\mathbb{N}}\times{\mathbb{N}}}$ denotes the identity matrix in ${\bf T}({\mathbb{C}}(t))$ and

[TABLE]

Now we need to invert the power series $e^{x}-c$ in ${\mathbb{C}}[[x]]$ for some $c\neq 1$. For the sake of convenience, let us replace the variable $x$ by $(c-1)y$, where $c\neq 1$. Our aim is to invert the power series $e^{(c-1)y}-c$ when $c\neq 1$. For this we recall the generating function for the Eulerian polynomials. The Eulerian polynomials $A_{n}(t)$’s are defined by the following exponential generating function

[TABLE]

Thus

[TABLE]

Hence

[TABLE]

This in turn yields that the inverse of ${\bf A_{2}}(f;t)$, which we denote by ${\bf B_{2}}(f;t)$, is given by the formula

[TABLE]

as $f(1)\neq 1$. It can be calculated that $A_{0}(t)=A_{1}(t)=1$. Hence

[TABLE]

Here also one cannot multiply both sides of (24) by ${\bf B_{2}}(f_{1};s_{1})$ to express the column vector ${\bf V}_{r}(f_{1},\ldots,f_{r};\ s_{1},\ldots,s_{r})$ in terms of ${\bf V}_{r-1}(g_{2},f_{3},\ldots,f_{r};\ s_{1}+s_{2},s_{3},\ldots,s_{r})$. This is not possible as the coefficients of the Eulerian polynomials grow very fast. In fact it is known that for each $n\geq 0$, the sum of the coefficients of $A_{n}(t)$ is $n!$.

To get around this difficulty we repeat our truncation process. For an integer $q\geq 1$, let $I=I_{q},J=J_{q}$ be as before. Then we rewrite (24) as follows:

[TABLE]

Now inverting ${\bf A_{2}}^{II}(f_{1};s_{1})$, we get

[TABLE]

where

[TABLE]

All the series of meromorphic functions involved in the products of matrices in formulas (27) and (28) converge normally on all compact subsets of ${\mathbb{C}}^{r}$. Moreover, all the entries of the matrices on the right hand side of (28) are holomorphic on the open set $U_{r}(q)$. Therefore the entries of ${\bf Z}^{I}(f_{1},\ldots,f_{r};\ s_{1},\ldots,s_{r})$ are also holomorphic in $U_{r}(q)$.

If we write $\pi_{q}(f_{1},\ldots,f_{r};\ s_{1},\ldots,s_{r})$ to be the first entry of ${\bf Z}^{I}(f_{1},\ldots,f_{r};\ s_{1},\ldots,s_{r})$, we then get from (27) that

[TABLE]

where $\pi_{q}(f_{1},\ldots,f_{r};\ s_{1},\ldots,s_{r})$ is holomorphic in the open set $U_{r}(q)$.

6. Poles and residues

In this section, using the formulas (21), (27) and the induction on depth $r$, we obtain a list of possible singularities of the multiple additive $L$-functions. From Remark 1, we know that when $r=1$ and $f_{1}(1)=1$, then $L_{1}(f_{1};s_{1})$ can be extended to a meromorphic function with only a simple pole at $s_{1}=1$ with residue $1$, and if $r=1$ but $f_{1}(1)\not=1$, then $L_{1}(f_{1};s_{1})$ can be extended to an entire function.

First we obtain an expression for the residues along the possible polar hyperplane of the multiple additive $L$-functions and then we deduce the exact set of singularities of these functions. A theorem of G. Frobenius [3] about the zeros of Eulerian polynomials plays a crucial role in this analysis.

6.1. Set of all possible singularities

The following theorem gives a description of possible singularities of the multiple additive $L$-functions.

Theorem 9**.**

The multiple additive $L$-function $L_{r}(f_{1},\ldots,f_{r};\ s_{1},\ldots,s_{r})$ has different set of singularities depending on the values of $g_{i}(1)$ for $1\leq i\leq r$.

$(a)$* If $g_{i}(1)\neq 1$ for all $1\leq i\leq r$, then $L_{r}(f_{1},\ldots,f_{r};\ s_{1},\ldots,s_{r})$ is a holomorphic function on ${\mathbb{C}}^{r}$.*

Now let $i_{1}<\cdots<i_{m}$ be all the indices such that $g_{i_{j}}(1)=1$ for all $1\leq j\leq m$. Then the set of all possible singularities of $L_{r}(f_{1},\ldots,f_{r};\ s_{1},\ldots,s_{r})$ is described as follows:

$(b)$* If $i_{1}=1$, then $L_{r}(f_{1},\ldots,f_{r};s_{1},\ldots,s_{r})$ is holomorphic outside the union of hyperplanes given by the equations*

[TABLE]

It has at most simple poles along each of these hyperplanes.

$(c)$* If $i_{1}\not=1$, then $L_{r}(f_{1},\ldots,f_{r};s_{1},\ldots,s_{r})$ is holomorphic outside the union of hyperplanes given by the equations*

[TABLE]

It has at most simple poles along each of these hyperplanes.

Proof.

We prove the assertions $(a),(b)$ and $(c)$ separately.

Proof of $(a)$: From Remark 1, we know that the assertion $(a)$ is true for depth $1$ multiple additive $L$-function. Now let $q\geq 1$ be an integer and $I=I_{q},~{}J=J_{q}$ be as before. We will make use of equation (27) for our proof.

The entries of the first row of the matrix ${\bf B_{2}}^{II}(f_{1};s_{1})$ are holomorphic on ${\mathbb{C}}^{r}$ and by the induction hypothesis, the entries of the column matrix ${\bf V}_{r-1}^{I}(g_{2},f_{3},\ldots,f_{r};\ s_{1}+s_{2},s_{3},\ldots,s_{r})$ are holomorphic on ${\mathbb{C}}^{r}$. Further the entries of the column vector ${\bf Z}^{I}(f_{1},\ldots,f_{r};\ s_{1},\ldots,s_{r})$ are holomorphic in $U_{r}(q)$. Since the open sets $U_{r}(q)$ for $q\geq 1$ cover ${\mathbb{C}}^{r}$, assertion $(a)$ follows.

Proof of $(b)$: For depth $1$ multiple additive $L$-function, the assertion $(b)$ follows from Remark 1. Again let $q\geq 1$ be an integer and $I=I_{q},~{}J=J_{q}$ be as defined earlier. We now complete the proof of assertion $(b)$ by making use of the equation (21).

The entries of the first row of the matrix ${\bf B_{1}}^{II}(s_{1}-1)$ are holomorphic outside the hyperplane given by the equation $s_{1}=1$ and have at most simple pole along this hyperplane. By the induction hypothesis, the entries of the column vector ${\bf V}_{r-1}^{I}(f_{2},\ldots,f_{r};\ s_{1}+s_{2}-1,s_{3},\ldots,s_{r})$ are holomorphic outside the union of the hyperplanes given by the equations

[TABLE]

and have at most simple poles along these hyperplanes. Finally the entries of the column vector ${\bf W}^{I}(f_{1},\ldots,f_{r};\ s_{1},\ldots,s_{r})$ are holomorphic in $U_{r}(q)$. Since ${\mathbb{C}}^{r}$ is covered by open sets of the form $U_{r}(q)$ for $q\geq 1$, assertion $(b)$ follows.

Proof of $(c)$: The proof of this assertion follows along the lines of the assertion $(a)$. The only difference is the induction hypothesis. Here the induction hypothesis implies that the entries of the column matrix ${\bf V}_{r-1}^{I}(g_{2},f_{3},\ldots,f_{r};\ s_{1}+s_{2},s_{3},\ldots,s_{r})$ are holomorphic outside the union of the hyperplanes given by the equations

[TABLE]

Now the proof of assertion $(c)$ follows mutatis mutandis the proof of assertion $(a)$. ∎

6.2. Expression for residues

Here we compute the residues of the multiple additive $L$-function of depth $r$ along the possible polar hyperplanes.

Theorem 10**.**

Let $i_{1}<\cdots<i_{m}$ be the indices such that $g_{i_{j}}(1)=1$ for all $1\leq j\leq m$. If $i_{1}=1$, then $L_{r}(f_{1},\ldots,f_{r};\ s_{1},\ldots,s_{r})$ has a polar singularity along the hyperplane given by the equation $s_{1}=1$ and the residue is the restriction of $L_{r-1}(f_{2},\ldots,f_{r};\ s_{2},\ldots,s_{r})$ to this hyperplane.

In general, the residue along the hyperplane given by the equation

[TABLE]

is the restriction of the product of $L_{r-i_{j}}(f_{i_{j}+1},\ldots,f_{r};\ s_{i_{j}+1},\ldots,s_{r})$ with $(0,j-n)$-th entry of

[TABLE]

to the above hyperplane.

Proof.

First suppose that $i_{1}=1$. Then for any integer $q\geq 1$, we deduce from (23) and Theorem 9 that

[TABLE]

has no pole inside the open set $U_{r}(q)$ along the hyperplane given by the equation $s_{1}=~{}1$. These open sets cover ${\mathbb{C}}^{r}$ and hence the residue of $L_{r}(f_{1},\ldots,f_{r};\ s_{1},\ldots,s_{r})$ along the hyperplane given by the equation $s_{1}=1$ is the restriction of the meromorphic function $L_{r-1}(f_{2},\ldots,f_{r};\ s_{2},\ldots,s_{r})$ to the hyperplane given by the equation $s_{1}=1$. This proves the first part of Theorem 10.

Next let $q\geq j-n+1$ be an integer and $I=I_{q},~{}J=J_{q}$ be as before. Now to determine the residue along the hyperplane

[TABLE]

we iterate the formulas (21) and (27) according to the applicable case and obtain that

[TABLE]

Here ${\bf Z}^{j,I}(f_{1},\ldots,f_{r};\ s_{1},\ldots,s_{r})$ is a column matrix whose entries are finite sums of products of rational functions in $s_{1},\ldots,s_{i_{j}-1}$ with meromorphic functions which are holomorphic in $U_{r}(q)$. These entries therefore have no singularity along the hyperplane given by the equation $s_{1}+\cdots+s_{i_{j}}=n$ in $U_{r}(q)$. The entries of ${\bf C}_{j}^{II}$ are rational functions in $s_{1},\ldots,s_{i_{j}-1}$ and hence again have no singularity along the above hyperplane. It now follows from the first part of Theorem 10 that the only entry of

[TABLE]

that can possibly have a pole in $U_{r}(q)$ along the hyperplane given by the equation $s_{1}+\cdots+s_{i_{j}}=n$ is the one of index $j-n$, which is

[TABLE]

Again by the first part of Theorem 10, the residue of this function along the hyperplane given by the equation $s_{1}+\cdots+s_{i_{j}}=n$ is the restriction of $L_{r-i_{j}}(f_{i_{j}+1},\ldots,f_{r};\ s_{i_{j}+1},\ldots,s_{r})$ to this hyperplane. Thus the residue of $L_{r}(f_{1},\ldots,f_{r};\ s_{1},\ldots,s_{r})$ along the hyperplane given by the equation $s_{1}+\cdots+s_{i_{j}}=n$ is the product of $L_{r-i_{j}}(f_{i_{j}+1},\ldots,f_{r};\ s_{i_{j}+1},\ldots,s_{r})$ with $(0,j-n)$-th entry of ${\bf C}_{j}^{II}$. Since open sets of the form $U_{r}(q)$ cover ${\mathbb{C}}^{r}$, this completes the proof of Theorem 10. ∎

6.3. Exact set of singularities

Now we will deduce the exact set of singularities of the multiple additive $L$-functions. As mentioned earlier, a theorem of Frobenius [3] about the zeros of Eulerian polynomials helps us in determining the exact set of singularities. We recall that the Eulerian polynomials $A_{n}(t)$’s are defined by the following exponential generating function

[TABLE]

These polynomials satisfy the following recurrence relation

[TABLE]

Frobenius [3] proved the following theorem about the zeros of the Eulerian polynomials $A_{n}(t)$.

Theorem 11** (Frobenius).**

All the zeros of the Eulerian polynomials $A_{n}(t)$ are real, negative and simple.

From this theorem and the above recurrence formula, we can now deduce the following corollary.

Corollary 1**.**

For any $n\geq 0$, $A_{n+1}(t)$ and $A_{n}(t)$ do not have a common zero.

Proof.

To see this, let $a$ be a zero of $A_{n}(t)$. Then by Theorem 11, $A_{n}^{\prime}(a)\neq 0$. Again by Theorem 11, $a\neq 0,1$. Using the above recurrence formula, we deduce that $A_{n+1}(a)\neq~{}0$. ∎

Recall that for an indeterminate $t$ and a non-trivial additive character $f:{\mathbb{Z}}\to{\mathbb{C}}^{*}$, we have

[TABLE]

and

[TABLE]

Clearly non-zero elements of each row and each column of these matrices are linearly independent as elements of ${\mathbb{C}}(t)$. Also the first two entries of the first row of these matrices are non-zero. Since we know that the Bernoulli numbers

[TABLE]

we get that at least one of the first two entries in every column of ${\bf B_{1}}(t)$ is non-zero. On the other hand, Corollary 1 implies that at least one of the first two entries in every column of ${\bf B_{2}}(f;t)$ is non-zero. Hence we know that for any two indeterminates $t_{1},t_{2}$ and any two non-trivial group homomorphisms $f_{1},f_{2}:{\mathbb{Z}}\to{\mathbb{C}}^{*}$, all entries of the first row of the matrices ${\bf B_{1}}(t_{1})\ {\bf B_{1}}(t_{2})$, ${\bf B_{1}}(t_{1})\ {\bf B_{2}}(f_{2};t_{2})$, ${\bf B_{2}}(f_{1};t_{1})\ {\bf B_{1}}(t_{2})$ and ${\bf B_{2}}(f_{1};t_{1})\ {\bf B_{2}}(f_{2};t_{2})$ are non-zero.

With this observation in place, we are now ready to determine the exact set of singularities of the multiple additive $L$-functions.

Theorem 12**.**

The exact set of polar singularities of the multiple additive $L$-function$L_{r}(f_{1},\ldots,f_{r};\ s_{1},\ldots,s_{r})$ of depth $r$ differs from the set of all possible singularities (as listed in Theorem 9) only in the following two cases. Here we keep the notations of Theorem 9.

a) If $i_{1}=1$ and $i_{2}=2$ i.e. $f_{1}(1)=f_{2}(1)=1$, then $L_{r}(f_{1},\ldots,f_{r};s_{1},\ldots,s_{r})$ is holomorphic outside the union of hyperplanes given by the equations

[TABLE]

where $J:=\{-2n-1:n\in{\mathbb{N}}\}$. It has simple poles along each of these hyperplanes.

b) If $i_{1}=2$, then $L_{r}(f_{1},\ldots,f_{r};s_{1},\ldots,s_{r})$ is holomorphic outside the union of hyperplanes given by the equations

[TABLE]

where $J:=\{1-n:n\in I\}$ and $I:=\{n\in{\mathbb{N}}:A_{n}(f_{1}(1))=0\}$. It has simple poles along each of these hyperplanes.

Proof.

When $1\leq j\leq m$ and $n\in{\mathbb{Z}}_{\leq j}$, the restriction of $L_{r-i_{j}}(f_{i_{j}+1},\ldots,f_{r};\ s_{i_{j}+1},\ldots,s_{r})$ to the hyperplane given by the equation $s_{1}+\cdots+s_{i_{j}}=n$ is a non-zero meromorphic function.

First suppose that $i_{1}=1$ and $i_{2}=2$. Then by (31), we deduce that only non-zero entries in the first row of ${\bf C}_{2}$ are of index $(0,1)$ and of index $(0,2n)$ for $n\in{\mathbb{N}}$. Also we know that all the entries in the first row of ${\bf C}_{j}$ for $3\leq j\leq m$ are non-zero. Now from Theorem 10, we conclude that the exact set of singularities in this case consists of the hyperplanes given by the equations

[TABLE]

where $J:=\{-2n-1:n\in{\mathbb{N}}\}$. This completes the proof of assertion $(a)$.

Next let $i_{1}=2$. Clearly the entries in the first row of ${\bf C}_{1}$ that are zero are of index $(0,n)$ for all $n\in I$. Also we know that all entries in the first row of ${\bf C}_{j}$ for $2\leq j\leq m$ are non-zero. Hence using Theorem 10, we have that the exact set of singularities in this case consists of the hyperplanes given by the equations

[TABLE]

where $J:=\{1-n:n\in I\}$.

Now in all other cases, applying Theorem 9, we see that the hyperplanes given by the equations of the form $s_{1}+s_{2}=n$ are not singularities of $L_{r}(f_{1},\ldots,f_{r};\ s_{1},\ldots,s_{r})$. Using Theorem 10, we know that the expression for residues along the possible polar hyperplanes given by the equations of the form $s_{1}+\cdots+s_{i_{j}}=n$ involves product of at least two matrices of the form ${\bf B_{1}}(x)$ and ${\bf B_{2}}(f;y)$ where $x$ and $y$ are two different indeterminates. Hence such an expression is non-zero. This completes the proof of Theorem 12. ∎

Example 1**.**

We know that $t=-1$ is a zero for the Eulerian polynomials $A_{n}(t)$ only when $n$ is even. Suppose that we are in a case when $f_{1}(1)=-1$ and $i_{1}=2$ i.e. $f_{1}(1)=f_{2}(1)=-1$. Now Theorem 12 implies that the exact set of singularities of $L_{r}(f_{1},\ldots,f_{r};\ s_{1},\ldots,s_{r})$ consists of the hyperplanes given by the equation

[TABLE]

where $J:=\{-2n-1:n\in{\mathbb{N}}\}$. Further $L_{r}(f_{1},\ldots,f_{r};\ s_{1},\ldots,s_{r})$ has simple poles along these hyperplanes.

7. Concluding remarks

Though at this moment it seems difficult to determine the exact set of singularities of the multiple Dirichlet $L$-functions, we can still extend the theorem of Akiyama and Ishikawa for Dirichlet characters, not necessarily of same modulus. As an immediate consequence of (10), Theorem 8 and Theorem 9, we derive the following theorem.

Theorem 13**.**

Let $r\geq 1$ be an integer and $\chi_{1},\ldots,\chi_{r}$ be Dirichlet characters of arbitrary modulus. Then the multiple Dirichlet $L$-function $L_{r}(s_{1},\ldots,s_{r};~{}\chi_{1},\ldots,\chi_{r})$ of depth $r$ can be extended as a meromorphic function to ${\mathbb{C}}^{r}$ with possible simple poles at the hyperplanes given by the equations

[TABLE]

It has been discussed in detail in [6] that how one could deduce translation formulas and thereafter obtain meromorphic continuation and information about singularities for various Dirichlet series and their several variable counterparts. In view of this, we conclude our article by the following question. An answer to this will perhaps help us to determine the exact set of singularities of the multiple Dirichlet $L$-functions.

Question**.**

Is it possible to derive a translation formula satisfied by the multiple Dirichlet $L$-functions of depth $r$, which involves its translates with respect to the first variable $s_{1}$ and also involves the multiple Dirichlet $L$-functions of depth $r-1$?

Acknowledgement: I would like to thank Prof. Joseph Oesterlé for his helpful suggestions and comments.