ON ANALYTIC CONTINUATION OF MULTIPLE L-FUNCTIONS AND RELATED ZETA-FUNCTIONS

SHIGEKI AKIYAMA & HIDEAKI ISHIKAWA

1. Introduction

Analytic continuation of Euler-Zagier’s multiple zeta function of two variables was first established by F.V.Atkinson [3] with an application to the mean value problem of the Riemann zeta function. We can find recent developments in [8],[7] and [5]. From an analytic point of view, these results suggest broad applications of multiple zeta functions. In [9] and [10], D.Zagier pointed out an interesting interplay between positive integer values and other areas of mathematics, which include knot theory and mathematical physics. Many works had been done according to his motivation but here we restrict our attention to the analytic continuation. T.Arakawa and M.Kaneko [2] showed an analytic continuation with respect to the last variable. To speak about the analytic continuation with respect to all variables, we have to refer to J. Zhao [11] and S.Akiyama, S.Egami and Y.Tanigawa [1]. In [11], an analytic continuation and the residue calculation were done by using the theory of generalized functions in the sense of I.M. Gel’fand and G.E. Shilov. In [1], they gave an analytic continuation by means of a simple application of the Euler-Maclaurin formula. The advantage of this method is that it gives the complete location of singularities. This work also includes some study on the values at non positive integers.

In this paper we consider a more general situation, which seems important for number theory, in light of the method of [1]. We shall give an analytic continuation of multiple Hurwitz zeta functions (Theorem 1) and also multiple L functions (Theorem 2) defined below. In special cases, we can completely describe the whole set of singularities, by using a property of zeros of Bernoulli polynomials (Lemma 4) and a non vanishing result on a certain character sum (Lemma 2).

We explain notations used in this paper. The rational integers is denoted by ℤ, the rational numbers by ℚ, the complex numbers by ℂ and the positive integers by ℕ. We write ℤ_{≤ℓ} for the integers not greater than ℓ. Let χ_{i} (i = 1, 2, …, k) be Dirichlet characters of the same conductor q ≥ 2 and β_{i} (i = 1, 2, ⋯, q) be real numbers in the half open interval [0, 1). The principal character is denoted by χ_{0}. Then multiple Hurwitz zeta function and multiple L function are defined respectively by:

ζ_{k}(s_{1}, …, s_{k} | β_{1}, …, β_{k}) = ∑_{0<n_{1}<···<n_{k}} \frac{1}{(n_{1} + β_{1})^{s_{1}}(n_{2} + β_{2})^{s_{2}} … (n_{k} + β_{k})^{s_{k}}} (1)

and

L_{k}(s_{1}, …, s_{k} | χ_{1}, …, χ_{k}) = ∑_{0<n_{1}<···<n_{k}} \frac{χ_{1}(n_{1})}{n_{1}^{s_{1}}} \frac{χ_{2}(n_{2})}{n_{2}^{s_{2}}} … \frac{χ_{k}(n_{k})}{n_{k}^{s_{k}}}, (2)

where n_{i} ∈ ℕ (i = 1, …, k). If ℜ(s_{i}) ≥ 1 (i = 1, 2, …, k − 1) and ℜ(s_{k}) > 1, then these series are absolutely convergent and define holomorphic functions of k complex variables in this region. In the sequel we write them by ζ_{k}(s | β) and L_{k}(s | χ), for abbreviation. The Hurwitz zeta function ζ(s, α) in the usual sense for α ∈ (0, 1) is written as

ζ(s, α) = ∑_{n=0}^{∞} \frac{1}{(n + α)^{s}} = \frac{1}{α^{s}} + ζ_{1}(s | α),

by the above notation.

We shall state the first result. Note that β_{j} − β_{j+1} = 1/2 for some j implies β_{j−1} − β_{j} ≠ 1/2, since β_{j} ∈ [0, 1).

Theorem 1. The multiple Hurwitz zeta function ζ_{k}(s | β) is meromorphically continued to ℂ^{k} and has possible singularities on:

s_{k} = 1,  ∑_{i=1}^{j} s_{k−i+1} ∈ ℤ_{≤j} (j = 2, 3, …, k).

Let us assume furthermore that all β_{i} (i = 1, …, k) are rational. If β_{k−1} − β_{k} is not 0 nor 1/2, then the above set coincides with the set of whole singularities. If β_{k−1} − β_{k} = 1/2 then

s_{k} = 1
s_{k−1} + s_{k} = 2, 0, −2, −4, −6, …
∑_{i=1}^{j} s_{k−i+1} ∈ ℤ_{≤j} for j = 3, 4, …, k

forms the set of whole singularities. If β_{k−1} − β_{k} = 0 then

s_{k} = 1
s_{k−1} + s_{k} = 2, 1, 0, −2, −4, −6, …
∑_{i=1}^{j} s_{k−i+1} ∈ ℤ_{≤j} for j = 3, 4, …, k

forms the set of whole singularities.

For the simplicity, we only concerned with special cases and determined the whole set of singularities in Theorem 1. The reader can easily handle the case when all β_{i} − β_{i+1} (i = 1, …, k − 1) are not necessary rational and fixed. So we have enough information on the location of singularities of multiple Hurwitz zeta functions. For the case of multiple L functions, our knowledge is rather restricted.

Theorem 2. The multiple L-function L_{k}(s|χ) is meromorphically continued to ℂ^{k} and has possible singularities on:

s_{k} = 1, ∑_{i=1}^{j} s_{k−i+1} ∈ ℤ_{≤j} (j = 2, 3, …, k).

Especially for the case k = 2, we can state the location of singularities in detail as follows:

Corollary 1. We have a meromorphic continuation of L_{2}(s | χ) to ℂ^{2}. L_{2}(s | χ) is holomorphic in

\begin{cases}\{(s_{1}, s_{2}) ∈ ℂ^{2} | s_{1} + s_{2} ∉ ℤ_{≤2}, s_{2} ≠ 1\} & if χ_{1} = χ_{0}, χ_{2} = χ_{0} \\ \{(s_{1}, s_{2}) ∈ ℂ^{2} | s_{1} + s_{2} ∉ ℤ_{≤1}, s_{2} ≠ 1\} & if χ_{1} ≠ χ_{0}, χ_{2} = χ_{0} \\ \{(s_{1}, s_{2}) ∈ ℂ^{2} | s_{1} + s_{2} ∉ ℤ_{≤1} \} & if χ_{2} ≠ χ_{0}, \end{cases} (3)

where the excluded sets are possible singularities. Suppose that χ_{1} and χ_{2} are primitive characters with χ_{1}χ_{2} ≠ χ_{0}. Then L_{2}(s | χ) is a holomorphic function in

\begin{cases}\{(s_{1}, s_{2}) ∈ ℂ^{2} | s_{1} + s_{2} ≠ 0, −2, −4, −6, −8, … \} & if χ_{1}χ_{2}(−1) = 1, \\ \{(s_{1}, s_{2}) ∈ ℂ^{2} | s_{1} + s_{2} ≠ 1, −1, −3, −5, −7, … \} & if χ_{1}χ_{2}(−1) = −1, \end{cases} (4)

where the excluded set forms the whole set of singularities.

Unfortunately the authors could not get the complete description of singularities of multiple L function for k ≥ 3.

2. Preliminaries

Let N_{1}, N_{2} ∈ ℕ and η be a real number. Suppose that a function f(x) is l + 1 times continuously differentiable. By using Stieltjes integral expression, we see

∑_{N_{1}+η<n≤N_{2}} f(n) = ∫_{N_{1}+η}^{N_{2}} f(x)d[x]

= ∫_{N_{1}+η}^{N_{2}} f(x)dx − [f(x)B̃_{1}(x)]_{N_{1}+η}^{N_{2}} + ∫_{N_{1}+η}^{N_{2}} f′(x)B̃_{1}(x)dx

where B̃_{j}(x) = B_{j}(x − [x]) is the j-th periodic Bernoulli polynomial. Here j-th Bernoulli polynomial B_{j}(x) is defined by

\frac{te^{xt}}{e^{t} − 1} = ∑_{j=0}^{∞} \frac{B_{j}(x)}{j!}t^{j}

and [x] is the largest integer not exceeding x. Define the Bernoulli number B_{r} by the value B_{r} = B_{r}(0). Repeating integration by parts,

∑_{N_{1}+η<n≤N_{2}} f(n) = ∫_{N_{1}+η}^{N_{2}} f(x)dx + \frac{1}{2}f(N_{2}) + f(N_{1} + η)B̃_{1}(η)

+ ∑_{r=1}^{l} \frac{(−1)^{r+1}}{(r + 1)!} (B_{r+1}f^{(r)}(N_{2}) − f^{(r)}(N_{1} + η)B̃_{r+1}(η)) (5)

−\frac{(−1)^{l+1}}{(l + 1)!} ∫_{N_{1}+η}^{N_{2}} f^{(l+1)}(x)B̃_{l+1}(x)dx.

When η = 0, the formula (5) is nothing but the standard Euler-Maclaurin summation formula. This slightly modified summation formula by a parameter η works quite fine in studying our series (1) and (2).

Lemma 1. Letting

Φ_{l}(s |N_{1} + η, α) = \frac{(s)_{l+1}}{(l + 1)!} ∫_{N_{1}+η}^{∞} \frac{B̃_{l+1}(x)}{(x + α)^{s+l+1}} dx

and

(s)_{r} = \begin{cases}s(s + 1)(s + 2) … (s + r − 1) & if r ≥ 1 \\ 1 & if r = 0 \\ (s − 1)^{−1} & if r = −1 \end{cases} ,

it follows that

∑_{N_{1}+η<n} \frac{1}{(n + α)^{s}} = ∑_{r=−1}^{l} \frac{B̃_{r+1}(η)}{(r + 1)!} \frac{(s)_{r}}{(N_{1} + α + η)^{s+r}} − Φ_{l}(s | N_{1} + η, α)

with

Φ_{l}(s | N_{1} + η, α) ≪ N_{1}^{−(ℜs+l+1)}.

Proof. Put f(x) = (x + α)^{−s}. Then we have f^{(r)}(x) = (−1)^{r}(s)_{r}(x + α)^{−s−r}. So from (5),

∑_{N_{1}+η<n}^{N_{2}} \frac{1}{(n + α)^{s}} = [\frac{1}{1 − s} \frac{1}{(x + α)^{s−1}} ]_{N_{1}+η}^{N_{2}} + \frac{1}{2} \frac{1}{(N_{2} + α)^{s}} + \frac{B̃_{1}(η)}{(N_{1} + η + α)^{s}}

− ∑_{r=1}^{l} \frac{(s)_{r}}{(r + 1)!} (\frac{B_{r+1}}{(N_{2} + α)^{s+r}} − \frac{B̃_{r+1}(η)}{(N_{1} + η + α)^{s+r}} )

−\frac{1}{(l + 1)!} ∫_{N_{1}+η}^{N_{2}} B̃_{l+1}(x)\frac{(s)_{l+1}}{(x + α)^{s+l+1}} dx.

When ℜs > 1, we have

∑_{N_{1}+η<n} \frac{1}{(n + α)^{s}} = \frac{1}{s − 1} \frac{1}{(N_{1} + α + η)^{s−1}} + \frac{B̃_{1}(η)}{(N_{1} + η + α)^{s}}

+ ∑_{r=1}^{l} \frac{B̃_{r+1}(η)}{(r + 1)!} \frac{(s)_{r}}{(N_{1} + α + η)^{s+r}} − Φ_{l}(s | N_{1} + η, α)

as N_{2} → ∞. When ℜs ≤ 1, if we take a sufficiently large l, the integral in the last term Φ_{l}(s | N_{1} + η, α) is absolutely convergent. Thus this formula gives an analytic continuation of the series of the left hand side. Performing integration by parts once more and comparing two expressions, it can be easily seen that Φ_{l}(s | N_{1} + η, α) ≪ N_{1}^{−(ℜs+l+1)}.

Let A_{χ_{1},χ_{2}}(j) be a sum

∑_{a_{1}=1}^{q−1} ∑_{a_{2}=1}^{q−1} χ_{1}(a_{1})χ_{2}(a_{2})B̃_{j}(\frac{a_{1} − a_{2}}{q}).

Lemma 2. Suppose χ_{1} and χ_{2} are primitive characters modulo q with χ_{1}χ_{2} ≠ χ_{0}. Then we have: for 1 ≤ j

A_{χ_{1},χ_{2}}(j) = \begin{cases}−2\frac{j!}{(2πi)^{j}} τ(χ_{1})\overline{τ(\overline{χ_{2}})}L(j, \overline{χ_{1}χ_{2}}) & if (−1)^{j}χ_{1}χ_{2}(−1) = 1, \\ 0 & if (−1)^{j}χ_{1}χ_{2}(−1) = −1, \end{cases}

where τ(χ) is the Gauss sum defined by τ(χ) = ∑_{u=0}^{q−1} χ(u)e^{2πiu/q}.

Proof. Recall the Fourier expansion of Bernoulli polynomial:

B̃_{j}(y) = −j! lim_{M→∞} ∑_{\begin{matrix}n=−M \\ n≠0 \end{matrix}}^{M} \frac{e^{2πiny}}{(2πin)^{j}} (6)

for 1 ≤ j, 0 ≤ y < 1 except (j, y) = (1, 0). First suppose j ≥ 2, then the right hand side of (6) is absolutely convergent. Thus it follows from (6) that

A_{χ_{1},χ_{2}}(j) = ∑_{a_{1}=1}^{q−1} ∑_{a_{2}=1}^{q−1} χ_{1}(a_{1})χ_{2}(a_{2}) (−j! ∑_{\begin{matrix}n=−∞ \\ n≠0 \end{matrix}}^{∞} \frac{exp[2πin\frac{a_{1}−a_{2}}{q}]}{(2πin)^{j}} ) .

Since

∑_{u=0}^{q−1} χ(u)e^{2πinu/q} = χ̄(n)τ(χ)

for a primitive character χ, we have

A_{χ_{1},χ_{2}}(j) = −j! ∑_{\begin{matrix}n≠0 \\ n=−∞ \end{matrix}}^{∞} \frac{χ̄_{1}(n)χ̄_{2}(−n)τ(χ_{1})τ(χ_{2})}{(2πin)^{j}},

from which the assertion follows immediately by the relation \overline{τ(\overline{χ})} = χ(−1)τ(χ). Next assume that j = 1. Dividing A_{χ_{1},χ_{2}}(1) into

A_{χ_{1},χ_{2}}(1) = ∑′ χ_{1}(a_{1})χ_{2}(a_{2})B̃_{1} (\frac{a_{1} − a_{2}}{q} ) + ∑_{a_{1}=1}^{q−1} χ_{1}χ_{2}(a_{1})B_{1} (7)

where ∑′ taken over all the terms 1 ≤ a_{1}, a_{2} ≤ q − 1 with a_{1} ≠ a_{2}. The second sum in (7) is equal to 0 by the assumption. By using (6), the first sum is

∑′ χ_{1}(a_{1})χ_{2}(a_{2})B̃_{1} (\frac{a_{1} − a_{2}}{q} )

= ∑′ χ_{1}(a_{1})χ_{2}(a_{2}) lim_{M→∞} (− ∑_{\begin{matrix}n=−M \\ n≠0 \end{matrix}}^{M} \frac{exp[2πin(a_{1} − a_{2})/q]}{2πin})

= − lim_{M→∞} ∑_{\begin{matrix}n=−M \\ n≠0 \end{matrix}}^{M} \frac{∑_{a_{1}=1}^{q−1} ∑_{a_{2}=1}^{q−1} χ_{1}(a_{1})χ_{2}(a_{2}) exp[2πin\frac{a_{1}−a_{2}}{q}] − ∑_{a_{1}=1}^{q−1} χ_{1}χ_{2}(a_{1})}{2πin}

Using ∑_{a_{1}=1}^{q−1} χ_{1}χ_{2}(a_{1}) = 0 again, we have

∑′ χ_{1}(a_{1})χ_{2}(a_{2})B̃_{1} (\frac{a_{1} − a_{2}}{q} ) = −\frac{1}{2πi} lim_{M→∞} ∑_{\begin{matrix}n=−M \\ n≠0 \end{matrix}}^{M} \frac{χ̄_{1}(n)χ̄_{2}(−n)τ(χ_{1})τ(χ_{2})}{n}.

Hence we get the result.

We recall the classical theorem of von Staudt & Clausen.

Lemma 3.

B_{2n} + ∑_{p−1 | 2n} \frac{1}{p} is an integer.

Here the summation is taken over all prime p such that p − 1 divides 2n.

Extending the former results of D.H.Lehmer and K.Inkeri, the distribution of zeros of Bernoulli polynomials is extensively studied in [4], where one can find a lot of references. On rational zeros, we quote here the result of [6].

Lemma 4. Rational zeros of Bernoulli polynomial B_{n}(x) must be 0, 1/2 or 1. These zeros occur when and only when in the following cases:

\begin{cases}B_{n}(0) = B_{n}(1) = 0 & n is odd n ≥ 3 \\ B_{n}(1/2) = 0 & n is odd n ≥ 1. \end{cases} (8)

We shall give its proof, for the convenience of the reader.

Proof. First we shall show that if B_{n}(γ) = 0 with γ ∈ ℚ then 2γ ∈ ℤ. The Bernoulli polynomial is explicitly written as

B_{n}(x) = ∑_{k=0}^{n} \binom{n}{k}B_{k}x^{n−k}.

Let γ = P/Q with P, Q ∈ ℤ and P, Q are coprime. Then we have

−\frac{P^{n}}{Q} = nB_{1}P^{n−1} + ∑_{k=2}^{n} \binom{n}{k}B_{k}P^{n−k}Q^{k−1}.

Assume that there exist a prime factor q ≥ 3 of Q. Then the right hand side is qintegral. Indeed, we see that B_{1} = −1/2 and qB_{k} is q-integral since the denominator of B_{k} is always square free, which is an easy consequence of Lemma 3. But the left hand side is not q-integral, we get a contradiction. This shows that Q must be a power of 2. Let Q = 2^{m} with a non negative integer m. Then we have

−\frac{P^{n}}{2^{m−1}} = −nP^{n−1} + ∑_{k=2}^{n} \binom{n}{k}B_{k}P^{n−k}2^{m(k−1)+1}.

If m ≥ 2 then we get a similar contradiction. Thus Q must divide 2, we see 2γ ∈ ℤ. Now our task is to study that values of Bernoulli polynomials at half integers. Since B_{0}(x) = 1 and B_{1}(x) = x − 1/2, the assertion is obvious if n < 2. Assume that n ≥ 2 and even. Then by Lemma 3, the denominator of B_{n} is divisible by 3. Recalling the relation

B_{n}(x + 1) − B_{n}(x) = nx^{n−1}, (9)

we have for any integer m

B_{n}(m) ≡ B_{n} (mod ℤ).

This shows B_{n}(m) ≠ 0. The relation

B_{n}(1/2) = (2^{1−n} − 1)B_{n} (10)

implies that

B_{n}(1/2) ≠ 0 (11)

for n ≥ 2 and even. We see that B_{n}(1/2) is not 3 integral from Lemma 3 and the relation (10). Combining (9), (11), we have for any integer m and any even integer n ≥ 2

B_{n}(1/2 + m) ≠ 0.

It is easy to show the assertion for the remaining case when n ≥ 2 is odd, by using (9) and (10).

3. Analytic continuation of multiple Hurwitz zeta functions

This section is devoted to the proof of Theorem 1. First we treat the double Hurwitz zeta function. By Lemma 1, we see

∑_{n_{1}+(β_{1}−β_{2})<n_{2}} \frac{1}{(n_{2} + β_{2})^{s_{2}}} =

∑_{r=−1}^{l} \frac{B̃_{r+1}(β_{1} − β_{2})}{(r + 1)!} \frac{(s_{2})_{r}}{(n_{1} + β_{1})^{s_{2}+r}} − Φ_{l}(s_{2} | n_{1} + β_{1} − β_{2}, β_{2}).. (12)

Suppose first that β_{1} ≥ β_{2}. Then the sum ∑_{n_{1}+β_{1}−β_{2}<n_{2}} means ∑_{n_{1}<n_{2}}, so it follows from (12) that

ζ_{2}(s | β) = ∑_{n_{1}=1}^{∞} \frac{1}{(n_{1} + β_{1})^{s_{1}}} ∑_{n_{1}<n_{2}} \frac{1}{(n_{2} + β_{2})^{s_{2}}}

= ∑_{n_{1}=1}^{∞} \frac{1}{(n_{1} + β_{1})^{s_{1}}} ∑_{n_{1}+(β_{1}−β_{2})<n_{2}} \frac{1}{(n_{2} + β_{2})^{s_{2}}}

= ∑_{n_{1}=1}^{∞} \frac{1}{(n_{1} + β_{1})^{s_{1}}}

× \{ ∑_{r=−1}^{l} \frac{B̃_{r+1}(β_{1} − β_{2})}{(r + 1)!} \frac{(s_{2})_{r}}{(n_{1} + β_{1})^{s_{2}+r}} − Φ_{l}(s_{2} | n_{1} + β_{1} − β_{2}, β_{2})\} .

Suppose that β_{1} < β_{2}. We consider

ζ_{2}(s | β) = ∑_{n_{1}=1}^{∞} \frac{1}{(n_{1} + β_{1})^{s_{1}}} \{\frac{−1}{(n_{1} + β_{2})^{s_{2}}} + ∑_{n_{1}≤n_{2}}\} . (13)

Noting that the sum ∑_{n_{1}+β_{1}−β_{2}<n_{2}} means ∑_{n_{1}≤n_{2}}, we apply (12) to the second term in the braces. For the first term in the braces, we use the binomial expansion:

\frac{1}{(n_{1} + β_{2})^{s_{2}}} = \frac{1}{(n_{1} + β_{1})^{s_{2}}} ( ∑_{m=0}^{v} \frac{(−1)^{m}(s_{2})_{m}}{m!} (\frac{β_{2} − β_{1}}{n_{1} + β_{1}} )^{m} + R_{v+1}) (14)

with R_{v+1} ≪ n_{1}^{v+1}. By applying (12) and (14) to (13), we have

ζ_{2}(s | β) = ∑_{n_{1}=1}^{∞} \frac{1}{(n_{1} + β_{1})^{s_{1}}} \{\frac{1}{s_{2} − 1} \frac{1}{(n_{1} + β_{1})^{s_{2}−1}}

+ ∑_{r=0}^{l} (\frac{B̃_{r+1}(β_{1} − β_{2})}{(r + 1)!} − \frac{(β_{1} − β_{2})^{r}}{r!} ) \frac{(s_{2})_{r}}{(n_{1} + β_{1})^{s_{2}+r}}

−Φ_{l}(s_{2} | n_{1} + β_{1} − β_{2}, β_{2}) − \frac{R_{l+1}}{(n_{1} + β_{1})^{s_{2}}} \} .

Recalling the relation (9) and combining the cases β_{1} ≤ β_{2} and β_{1} > β_{2}, we have

ζ_{2}(s | β) = \frac{1}{s_{2} − 1}ζ_{1}(s_{1} + s_{2} − 1 | β_{1})

+ ∑_{r=0}^{l} \frac{B_{r+1}(β_{1} − β_{2})}{(r + 1)!}(s_{2})_{r}ζ_{1}(s_{1} + s_{2} + r | β_{1}) (15)

− ∑_{n_{1}=1}^{∞} \frac{Φ_{l}^{∗}(s_{2}|n_{1} + β_{1} − β_{2}, β_{2})}{(n_{1} + β_{1})^{s_{1}}}

where

Φ_{l}^{∗}(s_{2}|n_{1} + β_{1} − β_{2}, β_{2}) = \begin{cases}Φ_{l}(s_{2}|n_{1} + β_{1} − β_{2}, β_{2}) & if β_{1} ≥ β_{2} \\ Φ_{l}(s_{2}|n_{1} + β_{1} − β_{2}, β_{2}) + \frac{R_{l+1}}{(n_{1}+β_{1})^{s_{2}}} & if β_{1} < β_{2} \end{cases}

The right hand side in (15) has meromorphic continuation except the last term. The last summation is absolutely convergent, and hence holomorphic, in ℜ(s_{1} +s_{2} +l) > 0. Thus we now have a meromorphic continuation to ℜ(s_{1} + s_{2} + l) > 0. Since we can choose arbitrary large l, we get a meromorphic continuation of ζ_{2}(s | β) to ℂ^{2}, holomorphic in

{(s_{1}, s_{2}) ∈ ℂ^{2} | s_{2} ≠ 1, s_{1} + s_{2} ≠ 2, 1, 0, −1, −2, −3, … } .

The exceptions in this set are the possible singularities occurring in (s_{2} − 1)^{−1} and

\frac{B_{r+1}(β_{1} − β_{2})}{(r + 1)!}(s_{2})_{r}ζ_{1}(s_{1} + s_{2} + r | β_{1}). (16)

Whether they are ‘real singularity’ or not depends on the choice of parameters β_{i} (i = 1, 2). For the case of multiple Hurwitz zeta functions with k variables,

ζ_{k}(s_{1}, …, s_{k} | β_{1}, …, β_{k}) =

∑_{n_{1}=1}^{∞} \frac{1}{(n_{1} + β_{1})^{s_{1}}} ∑_{n_{1}<n_{2}} \frac{1}{(n_{2} + β_{2})^{s_{2}}} …

⋯ ∑_{n_{k−2}<n_{k−1}} \frac{1}{(n_{k−1} + β_{k−1})^{s_{k−1}}} \{\frac{1}{s_{k} − 1} \frac{1}{(n_{k−1} + β_{k−1})^{s_{k}−1}}

+ ∑_{r=0}^{l} \frac{B_{r+1}(β_{k−1} − β_{k})}{(r + 1)!} \frac{(s_{k})_{r}}{(n_{k−1} + β_{k−1})^{s_{k}+r}} − Φ_{l}^{∗}(s_{k} | n_{k−1} + β_{k−1} − β_{k}, β_{k})\}

= \frac{1}{s_{k} − 1}ζ_{k−1}(s_{1}, …, s_{k−2}, s_{k−1} + s_{k} − 1 | β_{1}, …, β_{k−1})

+ ∑_{r=0}^{l} \frac{B_{r+1}(β_{k−1} − β_{k})}{(r + 1)!}(s_{k})_{r}ζ_{k−1}(s_{1}, …, s_{k−2}, s_{k−1} + s_{k} + r | β_{1}, …, β_{k−1})

− ∑_{0<n_{1}<n_{2}<···<n_{k−1}}^{∞} \frac{Φ_{l}^{∗}(s_{k} | n_{k−1} + β_{k−1} − β_{k}, β_{k})}{(n_{1} + β_{1})^{s_{1}} … (n_{k−1} + β_{k−1})^{s_{k−1}}} .

Since

∑_{0<n_{1}<···<n_{k−1}} \frac{Φ_{l}^{∗}(s_{k} | n_{k−1} + β_{k−1} − β_{k}, β_{k})}{(n_{1} + β_{1})^{s_{1}}(n_{2} + β_{2})^{s_{2}} … (n_{k−1} + β_{k−1})^{s_{k−1}}} ≪ ∑_{n_{k−1}} \frac{n_{k−1}^{−l−ℜ(s_{k})+k−3}}{n_{k−1}^{L}}

with L = ℜ(s_{k−1}) + ∑_{1≤j≤k−2,ℜ(s_{i})≤0} ℜ(s_{i}), the last summation is convergent absolutely in

l − k + 2 + ℜ(s_{k−1}) + ℜ(s_{k}) + ∑_{\begin{matrix}1≤j≤k−2 \\ ℜ(s_{i})≤0 \end{matrix}} ℜ(s_{i}) > 0.

Since l can be taken arbitrarily large, we get an analytic continuation of ζ_{k}(s | β) to ℂ^{k}. Now we study the set of singularities more precisely. The ‘singular part’ of ζ_{2}(s | β) is

\frac{ζ_{1}(s_{1} + s_{2} − 1 | β_{1})}{s_{2} − 1} + ∑_{r=0}^{∞} \frac{(s_{2})_{r}}{s_{1} + s_{2} + r − 1} \frac{B_{r+1}(β_{1} − β_{2})}{(r + 1)!}..

Note that this sum is by no means convergent and just indicates local singularities. From this expression we see

s_{2} = 1, s_{1} + s_{2} ∈ {2, 1, 0, −1, −2, −3, −4, … }

are possible singularities and the second assertion of the Theorem 1 for k = 2 is now clear with the help of Lemma 4. We wish to determine the whole singularities when all β_{i} (i = 1, …, k) are rational numbers by an induction on k. Let us consider the case of k variables,

ζ_{k}(s | β) =

∑_{r=−1}^{l} \frac{B_{r+1}(β_{k−1} − β_{k})}{(r + 1)!}(s_{k})_{r}ζ_{k−1}(s_{1}, …, s_{k−2}, s_{k−1} + s_{k} + r | β_{1}, … β_{k−1})

− ∑_{0<n_{1}<···<n_{k−1}} \frac{Φ_{l}^{∗}(s_{k} | n_{k−1} + β_{k−1} − β_{k}, β_{k})}{(n_{1} + β_{1})^{s_{1}} … (n_{k−1} + β_{k−1})^{s_{k−1}}} .

We shall only prove the case when β_{k−1} − β_{k} = 0. Other cases are left to the reader. By the induction hypothesis and Lemma 4 the singularities lie on, at least, for r = −1, 0, 1, 3, 5, 7, …,

s_{k} = 1, s_{k−1} + s_{k} + r = 1,
s_{k−2} + s_{k−1} + s_{k} + r = 2, 0, −2, −4, −6, …

and

s_{k−j} + s_{k−j+1} + ⋯ + s_{k} + r ∈ ℤ_{≤j}, for j ≥ 3,

for any three cases; β_{k−2} − β_{k−1} = 0, 1/2, and otherwise.

Thus

s_{k} = 1, s_{k−1} + s_{k} = 2, 1, 0, −2, −4, −6, …

and

s_{k−j+1} + s_{k−j+2} + ⋯ + s_{k} ∈ ℤ_{≤j}, for j ≥ 3

are the possible singularities, as desired. Note that the singularities of the form

s_{k−2} + s_{k−1} + s_{k} + r = 1, −1, −3, −5, …

may appear. However, these singularities don’t affect our description. Next we will show that they are the ’real’ singularities. For example, the singularities of the form s_{k−2} + s_{k−1} + s_{k} = η occurs in several ways for a fixed η. So our task is to show that no singularities defined by one of above equations will identically vanish in the summation process. This can be shown by a small trick of replacing variables:

s_{1} = u_{1}, …, s_{k−2} = u_{k−2}, s_{k−1} + s_{k} = u_{k−1}, s_{k} = u_{k}

In fact, we see that the singularities of ζ_{k}(u_{1}, …, u_{k−2}, u_{k−1} − u_{k}, u_{k} | β_{1}, …, β_{k}) appears in

∑_{r=−1}^{l} \frac{B_{r}(β_{k−1} − β_{k})}{(r + 1)!}(u_{k})_{r}ζ_{k−1}(u_{1}, …, u_{k−2}, u_{k−1} + r | β_{1}, … β_{k−1}).

By this expression we see that the singularities of ζ_{k−1}(u_{1}, …, u_{k−1}+r|β_{1}, … β_{k−1}) are summed with functions of u_{k} of different degree. Thus these singularities, as weighted sum by another variable u_{k}, will not vanish identically. This argument seems to be an advantage of [1], which clarify the exact location of singularities. The Theorem is proved by the induction.

4. Analytic continuation of multiple L-functions

Proof of Theorem 2. When ℜs_{i} > 1 for i = 1, 2, …, k, the series is absolutely convergent. Rearranging the terms,

∑_{n_{1}<···<n_{k}} \frac{χ_{1}(n_{1})}{n_{1}^{s_{1}}} … \frac{χ_{k}(n_{k})}{n_{k}^{s_{k}}} =

\frac{1}{q^{s_{1}+···+s_{k}}} ∑_{a_{1}=1}^{q−1} ∑_{a_{2}=1}^{q−1} ⋯ ∑_{a_{k}=1}^{q−1} χ_{1}(a_{1})χ_{2}(a_{2}) … χ_{k}(a_{k})

× [ ∑_{m_{1}=0}^{∞} \frac{1}{(m_{1} + \frac{a_{1}}{q})^{s_{1}}} ∑_{m_{1}+ \frac{a_{1}−a_{2}}{q} <m_{2}} \frac{1}{(m_{2} + \frac{a_{2}}{q})^{s_{2}}} …

⋯ ∑_{m_{k−2}+ \frac{a_{k−2}−a_{k−1}}{q} <m_{k−1}} \frac{1}{(m_{k−1} + \frac{a_{k−1}}{q})^{s_{k−1}}} ∑_{m_{k−1}+ \frac{a_{k−1}−a_{k}}{q} <m_{k}} \frac{1}{(m_{k} + \frac{a_{k}}{q})^{s_{k}}} ] .

By this expression, it suffices to show that the series in the last brace has the desirable property. When a_{i} − a_{i+1} ≥ 0 holds for i = 1, …, k − 1, this is clear form Theorem 1, since this series is just a multiple Hurwitz zeta function. Proceeding along the same line with the proof of Theorem 1, other cases are also easily deduced by recursive applications of Lemma 1. Since there are no need to use binomial expansions, this case is easier than before.

Proof of Corollary 1. Considering the case k = 2 in Theorem 2, we see

L_{2}(s | χ) = \frac{1}{q} \frac{∑_{a_{2}=1}^{q−1} χ_{2}(a_{2})}{s_{2} − 1}L(s_{1} + s_{2} − 1, χ_{1}) + \frac{1}{q^{s_{1}+s_{2}}} ∑_{a_{1}=1}^{q−1} ∑_{a_{2}=1}^{q−1} χ_{1}(a_{1})χ_{2}(a_{2})

× \{∑_{r=0}^{l} \frac{B̃_{r+1}(\frac{a_{1}−a_{2}}{q})}{(r + 1)!}(s_{2})_{r}ζ(s_{1} + s_{2} + r, \frac{a_{1}}{q}) − ∑_{m_{1}=0}^{∞} \frac{Φ_{l}(s_{2} | m_{1} + \frac{a_{1}−a_{2}}{q} , \frac{a_{2}}{q})}{(m_{1} + \frac{a_{1}}{q})^{s_{1}}}\} .

We have a meromorphic continuation of L_{2}(s | χ) to ℂ^{2}, which is holomorphic in the domain (3). Note that the singularities occur in

\frac{∑_{a_{2}=1}^{q−1} χ_{2}(a_{2})}{s_{2} − 1}L(s_{1} + s_{2} − 1, χ_{1})

and

∑_{a_{1}=1}^{q} ∑_{a_{2}=1}^{q} χ_{1}(a_{1})χ_{2}(a_{2})\frac{B̃_{r+1}(\frac{a_{1}−a_{2}}{q})}{(r + 1)!}(s_{2})_{r}ζ(s_{1} + s_{2} + r, \frac{a_{1}}{q}).

If χ_{2} is not principal then the first term vanishes and we see the ‘singular part’ is

∑_{r=0}^{∞} \frac{A_{χ_{1},χ_{2}}(r + 1)}{(r + 1)!} \frac{(s_{2})_{r}}{s_{1} + s_{2} + r − 1}.

Thus we get the result by using Lemma 2 and the fact:

L(n, χ) ≠ 0

for n ≥ 1 and a non principal character χ.

As we stated in the introduction, we do not have a satisfactory answer to the problem of describing whole sigularities of multiple L functions in the case k ≥ 3, at present. For example when k = 3, what we have to show is the non vanishing of the sum:

∑_{a_{1}=1}^{q−1} ∑_{a_{2}=1}^{q−1} ∑_{a_{3}=1}^{q−1} χ_{1}(a_{1})χ_{2}(a_{2})χ_{3}(a_{3})B̃_{r_{1}+1} (\frac{a_{1} − a_{2}}{q} ) B̃_{r_{2}+1} (\frac{a_{2} − a_{3}}{q} ) ,

apart from trivial cases.

References

[1] S. Akiyama, S. Egami, and Y. Tanigawa , An analytic continuation of multiple zeta functions and their values at non-positive integers, to appear in Acta Arith.

[2] T.Arakawa and M.Kaneko, Multiple zeta values, poly-Bernoulli numbers, and related zeta functions, Nagoya Math. J. 153 (1999), 189-209.

[3] F.V. Atkinson, The mean value of the Riemann zeta-function, Acta Math., 81 (1949), 353-376.

[4] K. Dilcher, Zero of Bernoulli, generalized Bernoulli and Euler polynomials, Memoirs of American Mathematical Society, number 386.

[5] S. Egami, Introduction to multiple zeta function, Lecture Note at Niigata University (in Japanese).

[6] K. Inkeri, The real roots of Bernoulli polynomials, Ann. Univ. Turku. Ser. A I 37 (1959), 20pp.

[7] M. Katsurada and K. Matsumoto, Asymptotic expansions of the mean values of Dirichlet L-functions. Math. Z., 208 (1991), 23–39.

[8] Y. Motohashi, A note on the mean value of the zeta and L-functions. I, Proc. Japan Acad., Ser. A Math. Sci. 61 (1985), 222–224.

[9] D. Zagier, Values of zeta functions and their applications, First European Congress of Mathematics, Vol. II, Birkhäuser, (1994) 210–220

[10] D. Zagier, Periods of modular forms, traces of Hecke operators, and multiple zeta values, Research into automorphic forms and L functions (in Japanese) ( Kyoto, 1992), Sūrikaisekikenkyūsho Kōkyūroku, 843 (1993), 162–170.

[11] J. Zhao, Analytic continuation of multiple zeta functions. Proc. Amer. Math. Soc. 128 (2000), no. 5, 1275–1283.

Shigeki AKIYAMA

Department of Mathematics, Faculty of Science, Niigata University, Ikarashi 2-8050, Niigata 950-2181, Japan e-mail: akiyama@math.sc.niigata-u.ac.jp

Hideaki ISHIKAWA

Graduate school of Natural Science, Niigata University, Ikarashi 2-8050, Niigata 950-2181, Japan e-mail: isikawah@qed.sc.niigata-u.ac.jp