Zeta Functions and Normalized Multiple Sine Functions

Shin-ya Koyama and Nobushige Kurokawa

Abstract. By using normalized multiple sine functions we show expressions for special values of zeta functions and L-functions containing ζ(3), ζ(5), etc. Our result reveals the importance of division values of normalized multiple sine functions. Properties of multiple Hurwitz zeta functions are crucial for the proof.

Mathematics Subject Classification 2000: 11M06

1 Introduction

The normalized sine function

S_{1}(x) = 2 sin(πx)

has the basic importance in number theory. This is expressed as

S_{1}(x) = Γ_{1}(x)^{−1}Γ_{1}(1 − x)^{−1}

with the normalized gamma function

Γ_{1}(x) = exp ( \frac{∂}{∂s}ζ(s, x)|_{s=0}) ,

where

ζ(s, x) = ∑_{n=0}^{∞}(n + x)^{−s}

is the Hurwitz zeta function. In fact, Lerch’s formula says that

Γ_{1}(x) = \frac{Γ(x)}{\sqrt{2π}}

for the usual gamma function Γ(x).

We know that the special value of S_{1}(x) at a rational number x ∈ ℚ with 0 < x < 1 is an algebraic integer

S_{1}(x) = (e^{−πix} − e^{πix}) i
= |1 − e^{2πix}| .

(Since S_{1}(1/3) = \sqrt{3}, the factor “2” is needed in S_{1}(x) to assure the integrality.) This algebraic integer is intimately related to the cyclotomic unit, and at the same time it appears in the socalled the class number formula of Dirichlet

L(1, χ) = −\frac{τ(χ)}{N} log ( ∏_{k=1}^{N−1} S_{1} (\frac{k}{N} )^{χ̄(k)}) ,

where

L(s, χ) = ∑_{n=1}^{∞} χ(n)n^{−s}

is the Dirichlet L-function for a non-trivial primitive even Dirichlet character χ modulo N and

τ(χ) = ∑_{k=1}^{N−1} χ(k)e^{2πik/N}

is the Gauss sum. We notice that the Dirichlet’s formula is written also as

L′(0, χ) = −\frac{1}{2} log ( ∏_{k=1}^{N−1} S_{1} (\frac{k}{N} )^{χ(k)})

via the functional equation.

The purpose of this paper is to generalize such a formula to the case of L(r, χ) for r ≥ 2 containing the Riemann zeta case χ = 1 by using the normalized multiple sine function S_{r}(x), which was constructed and studied in the previous paper [KK] (see §2 for a survey). We recall the construction. For ω_{1}, ..., ω_{r} > 0 and x > 0, the multiple Hurwitz zeta function is defined by Barnes [B] as

ζ_{r}(s, x; (ω_{1}, ..., ω_{r})) = ∑_{n_{1},...,n_{r}=0}^{∞}(n_{1}ω_{1} + ⋯ + n_{r}ω_{r} + x)^{−s}

in Re(s) > r. This has the analytic continuation to all s ∈ ℂ as a meromorphic function, and it is holomorphic at s = 0. Then the normalized multiple gamma function is defined as

Γ_{r}(x, (ω_{1}, ..., ω_{r})) = exp ( \frac{∂}{∂s}ζ_{r}(s, x; (ω_{1}, ..., ω_{r}))|_{s=0}) .

This is a constant multiple of multiple gamma function Γ_{r}^{B}(x; (ω_{1}, ..., ω_{r})) of Barnes [B]:

Γ_{r}(x; (ω_{1}, ..., ω_{r})) = Γ_{r}^{B}(x; (ω_{1}, ..., ω_{r}))/ρ_{r}(ω_{1}, ..., ω_{r}).

Now, the normalized multiple sine function is

S_{r}(x; (ω_{1}, ..., ω_{r})) = Γ_{r}(x; (ω_{1}, ..., ω_{r}))^{−1}Γ_{r}(ω_{1} + ⋯ + ω_{r} − x; (ω_{1}, ..., ω_{r}))^{(−1)^{r}}.

Hence, by the zeta regularized product (see Manin [M]), we can write

S_{r}(x; (ω_{1}, ..., ω_{r})) = ∏ ∐_{n_{1},...,n_{r}=0}^{∞}(n_{1}ω_{1} + ⋯ + n_{r}ω_{r} + x) ( ∐_{n_{1},...,n_{r}=1}^{∞} ∏(n_{1}ω_{1} + ⋯ + n_{r}ω_{r} − x))^{(−1)^{r−1}} .

For example

S_{1}(x, ω) = Γ_{1}(x, ω)^{−1}Γ_{1}(ω − x, ω)^{−1}
= ∏ ∐_{n=0}^{∞}(nω + x) ∏ ∐_{n=1}^{∞}(nω − x)
= 2 sin(πx/ω)

since we have

Γ_{1}(x, ω) = (2π)^{−1/2}Γ(x/ω)ω^{\frac{x}{ω} − \frac{1}{2}}

from

ζ_{1}(s, x, ω) = ω^{−s}ζ(s, x/ω).

To simplify the notation we put S_{r}(x) = S_{r}(x; (1, ..., 1)), Γ_{r}(x) = Γ_{r}(x; (1, ..., 1)) and ζ_{r}(s, x) = ζ_{r}(s, x; (1, ..., 1)). Hence

S_{r}(x) = Γ_{r}(x)^{−1}Γ_{r}(r − x)^{(−1)^{r}}

and

Γ_{r}(x) = exp ( \frac{∂}{∂s}ζ_{r}(s, x)|_{s=0}) .

This normalized multiple sine function S_{r}(x) has good properties similar to the usual sine function S_{1}(x) = 2 sin(πx). We refer to §2 for details. For example, it has the periodicity and the duplication formula:

S_{r}(x + 1) = S_{r}(x)S_{r−1}(x)^{−1}

and

S_{r}(2x) = ∏_{k=0}^{r} S_{r} (x + \frac{k}{2})^{\binom{r}{k}} .

Moreover S_{r}(x) satisfies the following differential equation:

\frac{S′_{r}(x)}{S_{r}(x)} = Q_{r}(x) cot πx

with Q_{r}(x) = (−1)^{r−1}π\binom{x−1}{r−1}. So, S_{r}(x) is a solution of the algebraic differential equation

S′′_{r}(x) + (πQ_{r}(x)^{−1} − 1) S′_{r}(x)^{2}S_{r}(x)^{−1} − Q′_{r}(x)Q_{r}(x)^{−1}S′_{r}(x) + πQ_{r}(x)S_{r}(x) = 0.

We also note that S_{r}(x) has the Weierstrass product expression similar to

S_{1}(x) = 2πx ∏_{n=1}^{∞} (1 − \frac{x^{2}}{n^{2}} )

= 2πx ∏_{n=1}^{∞} ((1 + \frac{x}{n}) ^{_{1}H_{n}} (1 − \frac{x}{n}) ^{_{1}H_{−n}}) .

For example

S_{2}(x) = 2πxe^{−x} ∏_{n=1}^{∞} ((1 + \frac{x}{n})^{n+1} (1 − \frac{x}{n})^{−n+1} e^{−2x})

= 2πxe^{−x} ∏_{n=1}^{∞} ((1 + \frac{x}{n}) ^{_{2}H_{n}} (1 − \frac{x}{n}) ^{_{2}H_{−n}} e^{−2x})

and

S_{3}(x) = 2πe^{−ζ′(−2)}xe^{\frac{x^{2}}{4} − \frac{3}{2} x} ∏_{n=1}^{∞} ((1 + \frac{x}{n})^{\frac{n^{2}}{2} + \frac{3n}{2} +1} (1 − \frac{x}{n})^{\frac{n^{2}}{2} − \frac{3n}{2} +1} e^{\frac{x^{2}}{2} −3x})

= 2πe^{−ζ′(−2)}xe^{\frac{x^{2}}{4} − \frac{3}{2} x} ∏_{n=1}^{∞} ((1 + \frac{x}{n}) ^{_{3}H_{n}} (1 − \frac{x}{n}) ^{_{3}H_{−n}} e^{\frac{x^{2}}{2} −3x})

(see §2).

Our main results are as follows. The first result expresses the values of the Riemann zeta function at positive odd integers.

Theorem 1.1 Let n = 1, 2, 3, ..., and for k = 1, 2, ..., n put

a(2n + 1, k) = ∑_{l=1}^{k}(−1)^{k−l}l^{2n}\binom{2n + 1}{k − l},

which is a positive integer. Then we have:

ζ′(−2n) = − log (∏_{k=1}^{n} S_{2n+1}(k)^{a(2n+1,k)}) .

(2)

ζ(2n + 1) = \frac{(−1)^{n−1}2^{2n+1}π^{2n}}{(2n)!} log (∏_{k=1}^{n} S_{2n+1}(k)^{a(2n+1,k)}) .

Examples 1.2 We have

ζ(3) = 4π^{2} log S_{3}(1),  (1.1)
ζ(5) = −\frac{4π^{4}}{3} log(S_{5}(1)S_{5}(2)^{11}),  (1.2)
ζ(7) = \frac{8π^{6}}{45} log(S_{7}(1)S_{7}(2)^{57}S_{7}(3)^{302}).  (1.3)

The above formula (1.1) was proved in [KK] previously. Remark 1.3 By the formula

S_{r}(k) = ∏_{l=0}^{k−1} S_{r−l}(1)^{\binom{k−1}{l}(−1)^{l}}  (1.4)

for 1 ≤ k < r (cf. §2), we can also express ζ(2n + 1) in terms of S_{l}(1) (2 ≤ l ≤ 2n + 1):

ζ(2n + 1) = \frac{(−1)^{n−1}2^{2n+1}π^{2n}}{(2n)!} log ( ∏_{l=2}^{2n+1} S_{l}(1)^{b(2n+1,l)})

with b(2n + 1, l) ∈ ℤ. Example 1.4 Since S_{5}(2) = S_{5}(1)S_{4}(1)^{−1} (see §2),

ζ(5) = −\frac{4π^{4}}{3} log(S_{5}(1)^{12}S_{4}(1)^{−11}).

Next, let χ be a non-trivial primitive Dirichlet character modulo N, and

L(s, χ) = ∑_{n=1}^{∞} χ(n)n^{−s}  (1.5)

the Dirichlet L-function. Then the values L(r, χ) for r = 1, 2, 3, ... are classified as

L(r, χ) = \{ \begin{matrix}π^{r} · (χ-Bernoulli number) & ⋯ & χ(−1) = (−1)^{r} \\ “difficult” & ⋯ & χ(−1) = (−1)^{r+1}. \end{matrix}

Here “difficult” means that these values have not been calculated explicitly yet except for the r = 1 case appearing in the Dirichlet’s class number formula.

We generalize Dirichlet’s result to some difficult case.

Theorem 1.5 Let χ be a primitive odd character modulo N. Then:

(1)

L′(−1, χ) = −\frac{1}{2} log ∏_{k=1}^{N−1} (S_{2} (\frac{k}{N} )^{N} S_{1} (\frac{k}{N} )^{k})^{χ(k)} .

(2)

L(2, χ) = \frac{2πiτ(χ)}{N^{2}} log ∏_{k=1}^{N−1} (S_{2} (\frac{k}{N} )^{N} S_{1} (\frac{k}{N} )^{k})^{χ̄(k)} .

Examples 1.6 We have

L(2, (\frac{−4}{∗})) = −\frac{π}{4} log (S_{2} (\frac{1}{4})^{4} S_{1} (\frac{1}{4}) S_{2} (\frac{3}{4})^{−4} S_{1} (\frac{3}{4})^{−3})

= \frac{π}{4} log (2^{3}S_{2}(\frac{1}{4})^{−8}) ,

L(2, (\frac{−3}{∗})) = −\frac{2\sqrt{3}π}{9} log (S_{2} (\frac{1}{3})^{3} S_{1} (\frac{1}{3}) S_{2} (\frac{2}{3})^{−3} S_{1} (\frac{2}{3})^{−2})

= \frac{4\sqrt{3}π}{9} log (3S_{2}(\frac{1}{3})^{−3}) ,

where we used the following relations (see §2):

S_{2}(1 − x) = S_{2}(1 + x)^{−1}
= (S_{2}(x)S_{1}(x)^{−1})^{−1}
= S_{2}(x)^{−1}S_{1}(x).

Theorem 1.7 Let χ be a non-trivial primitive even character modulo N. Then:

(1)

L′(−2, χ) = −\frac{1}{2} log ∏_{k=1}^{N−1} (S_{3} (\frac{k}{N} )^{2N^{2}} S_{2} (\frac{k}{N} )^{2Nk−3N^{2}} S_{1} (\frac{k}{N} )^{k^{2}})^{χ(k)} .

L(3, χ) = \frac{2π^{2}τ(χ)}{N^{3}} log ∏_{k=1}^{N−1} (S_{3} (\frac{k}{N} )^{2N^{2}} S_{2} (\frac{k}{N} )^{2Nk−3N^{2}} S_{1} (\frac{k}{N} )^{k^{2}})^{χ̄(k)} .

Example 1.8

L(3, (\frac{12}{∗})) = \frac{\sqrt{3}π^{2}}{432} log (S_{3}(\frac{1}{12})^{288}S_{2}(\frac{1}{12})^{−408}S_{1}(\frac{1}{12})S_{3}(\frac{5}{12})^{−288}S_{2}(\frac{5}{12})^{312}S_{1}(\frac{5}{12})^{−25}

S_{3}(\frac{7}{12})^{−288}S_{2}(\frac{7}{12})^{264}S_{1}(\frac{7}{12})^{−49}S_{3}(\frac{11}{12})^{288}S_{2}(\frac{11}{12})^{−164}S_{1}(\frac{11}{12})^{121}) .

Thus the values S_{r}(a) for a ∈ ℚ satisfying 0 < a < r are quite interesting in relation to zeta values. We formulate our expectation as

Expectation 1.9 S_{r}(a) ∈ \overline{ℚ} for a ∈ ℚ satisfying 0 < a < r.

The situation would become transparent when we generalize it as below:

Expectation 1.10 S_{r}(\frac{k_{1}ω_{1}+···+k_{r}ω_{r}}{N}; \underline{ω}) ∈ \overline{ℚ} for N = 1, 2, 3, ... and k_{i} = 0, 1, ..., N − 1.

It is easy to see that Expectation 1.9 is contained in Expectation 1.10 for \underline{ω} = (1, ..., 1), and Expectation 1.10 clearly indicates that we are studying division values of multiple sine functions.

We note that Shintani [Sh] deeply studied S_{2}(x, (1, ε)) for a fundamental unit ε of a real quadratic field. In particular, he showed its appearance in the expression for a special value of a suitable L-function, and he obtained certain algebraicity such as

S_{2} (\frac{1}{3}, (1, ε)) S_{2} (1 + \frac{ε}{3}, (1, ε)) S_{2} (\frac{2 + 2ε}{3}, (1, ε)) = \sqrt{\frac{\frac{1+\sqrt{21}}{2} − \sqrt{\frac{3+\sqrt{21}}{2}}}{2}}

for ε = \frac{5+\sqrt{21}}{2}, which is the fundamental unit of ℚ(\sqrt{21}). Moreover, Shintani studied Kronecker’s Jugendtraum for a real quadratic field by using S_{2}(x, (1, ε)) (he denoted it by F (x; (1, ε))^{−1}). It might be valuable to report the following general product formula

∏_{{k_{1},...,kr=0 \atop (k_{1},...,kr)≠(0,...,0)}}^{N−1} S_{r} (\frac{k_{1}ω_{1} + ⋯ + k_{r}ω_{r}}{N}; (ω_{1}, ..., ω_{r})) = N

for an integer N ≥ 2. (See §2.)

Theorem 1.11 (1) Expectations 1.9 and 1.10 are valid for r = 1.

(2) Expectations 1.9 and 1.10 are valid for r = 2 with N = 2. Actually

S_{2} (\frac{ω_{1}}{2} ; \underline{ω}) = S_{2} (\frac{ω_{2}}{2} ; \underline{ω}) = \sqrt{2}

and

S_{2} (\frac{ω_{1} + ω_{2}}{2} ; \underline{ω}) = 1.

Remark 1.12 This paper was referred to in [KK] as a preprint in 2001.

2 Multiple Sine functions

The basic properties of multiple sine functions were proved in [K] and [KK]. Here we recall some of them. Theorem 2.1 [KK, Theorem 2.1] The multiple sine function S_{r}(x, \underline{ω}) satisfies the following identities:

(a) For \underline{ω} = (ω_{1}, ..., ω_{r}) ∈ ℝ_{+}^{r} put \underline{ω}(i) = (ω_{1}, ..., ω_{i−1}, ω_{i+1}, ..., ω_{r}) ∈ ℝ_{+}^{r−1}, then we have

S_{r}(x + ω_{i}, \underline{ω}) = S_{r}(x, \underline{ω})S_{r−1}(x, \underline{ω}(i))^{−1},  (2.1)

where we put S_{0}(x, ·) ≡ −1.

(b) For a positive integer N, we have

S_{r}(Nx, \underline{ω}) = ∏_{0≤k_{i}≤N−1} S_{r} (x + \frac{k · \underline{ω}}{N} , \underline{ω}) ,  (2.2)

where the product is taken over the vectors k = (k_{1}, ..., k_{r}). (c)

∏_{{0≤k_{i}≤N−1 \atop k≠0}} S_{r} (\frac{k · \underline{ω}}{N} , \underline{ω}) = N.

(d)

S_{r}(0, \underline{ω}) = 0.

(e) We have for any c > 0 the homogeneity S_{r}(cx, c\underline{ω}) = S_{r}(x, \underline{ω}).

Theorem 2.2 (a) For r ≥ 2 we have

S_{r}(x + 1) = S_{r}(x)S_{r−1}(x)^{−1}.

(b)

S_{r}(2x) = ∏_{k=0}^{r} S_{r} (x + \frac{k}{2})^{\binom{r}{k}} .

(c) Put Q_{r}(x) = (−1)^{r−1}π\binom{x−1}{r−1}, then

\frac{S′_{r}(x)}{S_{r}(x)} = Q_{r}(x) cot (πx).

(d)

S′′_{r}(x) + (πQ_{r}(x)^{−1} − 1) S′_{r}(x)^{2}S_{r}(x)^{−1} − Q′_{r}(x)Q_{r}(x)^{−1}S′_{r}(x) + πQ_{r}(x)S_{r}(x) = 0.

(e)

S_{2}(x) = 2πxe^{−x} ∏_{n=1}^{∞} ((1 + \frac{x}{n})^{n+1} (1 − \frac{x}{n})^{−n+1} e^{−2x})

= 2πxe^{−x} ∏_{n=1}^{∞} ((1 + \frac{x}{n}) ^{_{2}H_{n}} (1 − \frac{x}{n}) ^{_{2}H_{−n}} e^{−2x}) .

(f)

S_{3}(x) = 2πe^{−ζ′(−2)}xe^{\frac{x^{2}}{4} − \frac{3}{2} x} ∏_{n=1}^{∞} ((1 + \frac{x}{n})^{\frac{n^{2}}{2} + \frac{3n}{2} +1} (1 − \frac{x}{n})^{\frac{n^{2}}{2} − \frac{3n}{2} +1} e^{\frac{x^{2}}{2} −3x})

= 2πe^{−ζ′(−2)}xe^{\frac{x^{2}}{4} − \frac{3}{2} x} ∏_{n=1}^{∞} ((1 + \frac{x}{n}) ^{_{3}H_{n}} (1 − \frac{x}{n}) ^{_{3}H_{−n}} e^{\frac{x^{2}}{2} −3x}) .

Proof. The assertions (a) and (b) are immediate consequences from [KK, Theorem 2.1]. The differential equation (c) is proved in [KK, Theorem 2.15]. We compute from (c) that

(Q_{r}(x)^{−1} \frac{S′_{r}}{S_{r}} (x))′ = (cot πx)′
= −\frac{π}{sin^{2} πx}
= −π(cot^{2}(πx) + 1)
= −π ((Q_{r}(x)^{−1} \frac{S′_{r}}{S_{r}} (x))^{2} + 1) ,

which gives the proof of (d). Finally (e) and (f) are deduced from [KK, Examples 3.6], where we express the normalized multiple sine functions S_{r}(x) in terms of primitive multiple sine functions which are defined by the Hadamard product.

3 The Riemann zeta function

Lemma 3.1 There exist uniquely determined integers a(r, k) such that

x^{r−1} = ∑_{k=1}^{r−1} a(r, k) _{r}H_{x−k}  (3.1)

with _{r}H_{x−k} = \frac{(x−k+r−1)···(x−k+1)}{(r−1)!} for an indeterminate x. Indeed a(r, k) are given as follows:

a(r, k) = ∑_{l=1}^{k}(−1)^{k−l}l^{r−1}\binom{r}{k − l}.  (3.2)

Moreover,

a(r, r − k) = a(r, k).  (3.3)

Proof. The existence of a(r, k) follows from the fact that the (r − 1) polynomials _{r}H_{x−k} (k = 1, ..., r − 1) are linearly independent over ℚ. By putting x = k in (3.1), we have

k^{r−1} = a(r, 1)\binom{k + r − 2}{r − 1} + a(r, 2)\binom{k + r − 3}{r − 1} + ⋯ + a(r, k) · 1.

This leads to

a(r, k) = k^{r−1} − ∑_{j=1}^{k−1} a(r, j)\binom{k + r − 1 − j}{r − 1}.

Thus (3.2) is proved by induction on k. Next, from (3.1)

(−x)^{r−1} = ∑_{k=1}^{r−1} a(r, k) _{r}H_{−x−k}

and

rH_{−x−k} = \frac{(−x − k + r − 1) ⋯ (−x − k + 1)}{(r − 1)!} = (−1)^{r−1} _{r}H_{x−(r−k)},

so

x^{r−1} = ∑_{k=1}^{r−1} a(r, k) _{r}H_{x−(r−k)} = ∑_{k=1}^{r−1} a(r, r − k) _{r}H_{x−k}.

Hence, by the uniquenes of a(r, k) we have a(r, r − k) = a(r, k).

Examples 3.2 For x = n ∈ ℤ and r = 2, 3, 4, 5 we have

n = _{2}H_{n−1},
n^{2} = _{3}H_{n−1} + _{3}H_{n−2},
n^{3} = _{4}H_{n−1} + 4 _{4}H_{n−2} + _{4}H_{n−3},
n^{4} = _{5}H_{n−1} + 11 _{5}H_{n−2} + 11 _{5}H_{n−3} + _{5}H_{n−4}.

Proof of Theorem 1.1:

For r ≥ 2 we have by Lemma 3.1

ζ(s + 1 − r) = ∑_{n=1}^{∞} \frac{n^{r−1}}{n^{s}}

= ∑_{k=1}^{r−1} a(r, k) ∑_{n=1}^{∞} \frac{_{r}H_{n−k}}{n^{s}}

= ∑_{k=1}^{r−1} a(r, k)ζ_{r}(s, k),

where ζ_{r}(s, k) is the multiple Hurwitz zeta function

ζ_{r}(s, k) = ∑_{n=0}^{∞} \frac{_{r}H_{n}}{(n + k)^{s}} .

Thus we have

ζ′(1 − r) = ∑_{k=1}^{r−1} a(r, k) log Γ_{r}(k).

In case r = 2n + 1, it follows that

ζ′(−2n) = ∑_{k=1}^{2n} a(2n + 1, k) log Γ_{2n+1}(k)

= − ∑_{k=1}^{n} a(2n + 1, k) log S_{2n+1}(k)

= − log (∏_{k=1}^{n} S_{2n+1}(k)^{a(2n+1,k)}) ,

where we used S_{2n+1}(k) = Γ_{2n+1}(k)^{−1}Γ_{2n+1}(2n + 1 − k)^{−1} and a(2n + 1, 2n + 1 − k) = a(2n + 1, k).

Examples 3.3 We saw in [KK, Theorem 3.8(c)] that

ζ(3) = 4π^{2} log S_{3}(1).

Combining this with the fact that

S_{3}(1) = \sqrt{2}S_{3} (\frac{1}{2})^{−4/3} ,

which can be obtained by the facts

S_{3}(1) = S_{3}(2 · \frac{1}{2})
= S_{3}(\frac{1}{2})S_{3}(1)^{3}S_{3}(\frac{3}{2})^{3}S_{3}(2)
= S_{3}(1)^{4}S_{3}(\frac{1}{2})^{4}S_{2}(\frac{1}{2})^{−3}

and that S_{2}(\frac{1}{2}) = \sqrt{2}, we have

ζ(3) = \frac{16π^{2}}{3} log (S_{3} (\frac{1}{2})^{−1} 2^{\frac{3}{8}} )

which was proved in [KK, Theorem 3.8(b)] by another method (using a “primitive multiple sine function”).

4 Dirichlet L-functions for odd characters

We prove the formula for L(2, χ) for odd characters. Since our method follows a proof for Dirichlet’s result on L(1, χ) for even characters, we first recall it. We show the formula for L′(0, χ). Then the result on L(1, χ) follows via the functional equation.

Let χ be a non-trivial primitive Dirichlet character modulo N. We have

L(s, χ) = ∑_{k=1}^{N−1} χ(k) ∑_{m=0}^{∞} \frac{1}{(mN + k)^{s}}

= N^{−s} ∑_{k=1}^{N−1} χ(k)ζ(s, \frac{k}{N}),

where

ζ(s, x) = ∑_{m=0}^{∞} \frac{1}{(m + x)^{s}}

is the Hurwitz zeta function. Hence

L(0, χ) = ∑_{k=1}^{N−1} χ(k)ζ(0, \frac{k}{N})

and

L′(0, χ) = ∑_{k=1}^{N−1} χ(k)ζ′(0, \frac{k}{N}) − (log N) ∑_{k=1}^{N−1} χ(k)ζ(0, \frac{k}{N})
= ∑_{k=1}^{N−1} χ(k)ζ′(0, \frac{k}{N}) − (log N)L(0, χ).

When χ is even, it holds that L(0, χ) = 0 (this is the reason of “difficult”), so we have

L′(0, χ) = ∑_{k=1}^{N−1} χ(k)ζ′(0, \frac{k}{N})
= ∑_{k=1}^{N−1} χ(k) log Γ_{1} (\frac{k}{N} )
= \frac{1}{2} ∑_{k=1}^{N−1} χ(k) (log Γ_{1} (\frac{k}{N} ) + log Γ_{1} (\frac{N − k}{N} ))
= −\frac{1}{2} ∑_{k=1}^{N−1} χ(k) log S_{1} (\frac{k}{N} ) .

This gives the Dirichlet’s result. Proof of Theorem 1.5

We prove (1), then (2) is obtained via the functional equation. Since

ζ(s − 1, x) = ∑_{n=0}^{∞} \frac{n + x}{(n + x)^{s}} = ∑_{n=0}^{∞} \frac{n + 1}{(n + x)^{s}} + (x − 1) ∑_{n=0}^{∞} \frac{1}{(n + x)^{s}} = ζ_{2}(s, x) + (x − 1)ζ_{1}(s, x),

we have

ζ′(−1, x) = ζ′_{2}(0, x) + (x − 1)ζ′_{1}(0, x),

as ζ_{1}(s, x) = ζ(s, x). Now that χ is odd and that L(−1, χ) = 0, we compute

L′(−1, χ) = N ∑_{k=1}^{N−1} χ(k)ζ′(−1, \frac{k}{N})
= N ∑_{k=1}^{N−1} χ(k)ζ′_{2}(0, \frac{k}{N}) + N ∑_{k=1}^{N−1} χ(k)(\frac{k}{N} − 1)ζ′_{1}(0, \frac{k}{N})
= N ∑_{k=1}^{N−1} χ(k) log Γ_{2}(\frac{k}{N}) + N ∑_{k=1}^{N−1} χ(k)(\frac{k}{N} − 1) log Γ′_{1}(\frac{k}{N})
= N ∑_{k=1}^{N−1} χ(k) log (Γ_{2}(\frac{k}{N})Γ_{1}(\frac{k}{N})^{\frac{k}{N} −1})
= \frac{N}{2} ∑_{k=1}^{N−1} χ(k) log (\frac{Γ_{2}(\frac{k}{N})}{Γ_{2}(1 − \frac{k}{N})} \frac{Γ_{1}(\frac{k}{N})^{\frac{k}{N} −1}}{Γ_{1}(1 − \frac{k}{N})^{− \frac{k}{N}}} )
= \frac{N}{2} ∑_{k=1}^{N−1} χ(k) log (\frac{Γ_{2}(\frac{k}{N})}{Γ_{2}(2 − \frac{k}{N})} (Γ_{1}(\frac{k}{N})Γ_{1}(1 − \frac{k}{N}))^{\frac{k}{N} −1})
= −\frac{N}{2} ∑_{k=1}^{N−1} χ(k) log (S_{2}(\frac{k}{N})S_{1}(\frac{k}{N})^{\frac{k}{N} −1})
= −\frac{N}{2} ∑_{k=1}^{N−1} χ(k) log (S_{2}(\frac{k}{N})S_{1}(\frac{k}{N})^{\frac{k}{N}} ) ,

where we used the fact S_{1}(\frac{k}{N}) = S_{1}(\frac{N−k}{N}) with χ(N − k) = −χ(k).

5 Dirichlet L-functions for even characters

Proof of Theorem 1.7

We again show (1), then (2) is obtained via the functional equation. Since

(n + x)^{2} = 2 _{3}H_{n} + (2x − 3) _{2}H_{n} + (x − 1)^{2} _{1}H_{n},

we have

ζ(s − 2, x) = ∑_{n=0}^{∞} \frac{(n + x)^{2}}{(n + x)^{s}} = 2ζ_{3}(s, x) + (2x − 3)ζ_{2}(s, x) + (x − 1)^{2}ζ_{1}(s, x).

Therefore we have ζ′(−2, x) = 2ζ′_{3}(0, x) + (2x − 3)ζ′_{2}(0, x) + (x − 1)^{2}ζ′_{1}(0, x).

Now that χ is even and that L(−2, χ) = 0, we compute

L′(−2, χ) = N^{2} ∑_{k=1}^{N−1} χ(k)ζ′(−2, \frac{k}{N})

= N^{2} ∑_{k=1}^{N−1} χ(k) (2ζ′_{3}(0, \frac{k}{N}) + (2\frac{k}{N} − 3)ζ′_{2}(0, \frac{k}{N}) + (\frac{k}{N} − 1)^{2}ζ′_{1}(0, \frac{k}{N}))

= N^{2} ∑_{k=1}^{N−1} χ(k) log (Γ_{3}(\frac{k}{N})^{2}Γ_{2}(\frac{k}{N})^{2 \frac{k}{N} −3}Γ_{1}(\frac{k}{N})^{( \frac{k}{N} −1)^{2}})

= −\frac{1}{2} log ∏_{k=1}^{N−1} (S_{3}(\frac{k}{N})^{2N^{2}}S_{2}(\frac{k}{N})^{2Nk−3N^{2}}S_{1}(\frac{k}{N})^{k^{2}})^{χ(k)} .

6 Division values of normalized multiple sines

Proof of Theorem 1.11 Since S_{1}(x, ω) = 2 sin(\frac{πx}{ω}) by [KK, §2], we have

S_{1}(\frac{kω}{N}, ω) = 2 sin(\frac{kπ}{N}) = −i(e^{iπk/N} − e^{−iπk/N}) ∈ \overline{ℚ},

which leads to (1).

Recall that

S_{2}(x, (ω_{1}, ω_{2})) = \frac{Γ_{2}(ω_{1} + ω_{2} − x, (ω_{1}, ω_{2}))}{Γ_{2}(x, (ω_{1}, ω_{2}))}.

First

S_{2}(\frac{ω_{1}+ω_{2}}{2} , (ω_{1}, ω_{2})) = \frac{Γ_{2}(\frac{ω_{1}+ω_{2}}{2} , (ω_{1}, ω_{2}))}{Γ_{2}(\frac{ω_{1}+ω_{2}}{2} , (ω_{1}, ω_{2}))} = 1.

Secondly

S_{2}(\frac{ω_{1}}{2} , (ω_{1}, ω_{2})) = \frac{Γ_{2}(\frac{ω_{1}}{2} + ω_{2}, (ω_{1}, ω_{2}))}{Γ_{2}(\frac{ω_{1}}{2} , (ω_{1}, ω_{2}))}.

Here we use ([KK, §2])

Γ_{2}(x + ω_{2}, (ω_{1}, ω_{2})) = Γ_{2}(x, (ω_{1}, ω_{2}))Γ_{1}(x, ω_{1})^{−1}.

Then

Γ_{2}(\frac{ω_{1}}{2} + ω_{2}, (ω_{1}, ω_{2})) = Γ_{2}(\frac{ω_{1}}{2} , (ω_{1}, ω_{2}))Γ_{1}(\frac{ω_{1}}{2} , ω_{1})^{−1}.

Hence

S_{2}(\frac{ω_{1}}{2} , (ω_{1}, ω_{2})) = Γ_{1}(\frac{ω_{1}}{2} , ω_{1})^{−1}.

Now ([KK, §2])

Γ_{1}(x, ω) = \frac{Γ(\frac{x}{ω})}{\sqrt{2π}}ω^{\frac{x}{ω} − \frac{1}{2}} ,

so

Γ_{1}(\frac{ω_{1}}{2} , ω_{1}) = \frac{Γ(\frac{1}{2})}{\sqrt{2π}} = \frac{1}{\sqrt{2}}.

Thus

S_{2}(\frac{ω_{1}}{2} , (ω_{1}, ω_{2})) = \sqrt{2}.

Remark 6.1 A suitable restriction on the form of division points such as made in Expectation 1.10 will be needed as the following example shows:

S_{2}(2, (1, \sqrt{2})) ∉ \overline{ℚ}.  (6.1)

By this example, we must seriously look at S_{r}(a_{1}ω_{1} + ⋯ + a_{r}ω_{r}; (ω_{1}, ⋯, ω_{r})) for general a_{i} ∈ ℚ. The proof of (6.1) is given by

\frac{S_{2}(2, (1, \sqrt{2}))}{S_{2}(1, (1, \sqrt{2}))} = \frac{S_{2}(1 + 1, (1, \sqrt{2}))}{S_{2}(1, (1, \sqrt{2}))} = S_{1}(1, \sqrt{2})^{−1} = (2 sin \frac{π}{\sqrt{2}})^{−1} ∉ \overline{ℚ},

where

2 sin \frac{π}{\sqrt{2}} = −i(e^{i \frac{π}{\sqrt{2}}} − e^{−i \frac{π}{\sqrt{2}}} )
= −i((−1)^{1/\sqrt{2}} − ((−1)^{1/\sqrt{2}})^{−1})

and we used the transcendency result of Gelfond-Schneider (−1)^{1/\sqrt{2}} ∉ \overline{ℚ}. Moreover we appeal to the following facts:

S_{2}(ω_{1}, (ω_{1}, ω_{2})) = \sqrt{\frac{ω_{2}}{ω_{1}}} ,
S_{2}(ω_{2}, (ω_{1}, ω_{2})) = \sqrt{\frac{ω_{1}}{ω_{2}}} .

In particular

S_{2}(1, (1, \sqrt{2})) = 2^{\frac{1}{4}} ∈ \overline{ℚ}.

Thus we obtain (6.1).

References

[B] E.W. Barnes: On the theory of the multiple gamma function. Trans. Cambridge Philos. Soc., 19 (1904) 374-425.

[K] N. Kurokawa: Gamma factors and Plancherel measures. Proc. Japan Acad. 68A (1992) 256-260.

[KK] N. Kurokawa and S. Koyama: Multiple sine functions. Forum Math. 15 (2003) 839-876.

[KW] N. Kurokawa and M. Wakayama: On ζ(3). J. Ramanujan Math. Soc. 16 (2001) 205-214.

[M] Yu. I. Manin: Lectures on zeta functions and motives (according to Deninger and Kurokawa). Asterisque 228 (1995) 121-163.

[Sh] T. Shintani: On a Kronecker limit formula for real quadratic fields. J. Fac. Sci. Univ. Tokyo, 24 (1977) 167-199.

S. Koyama: 2-5-27, Hayabuchi, Tsuzuki-ku, Yokohama, 224-0025, Japan (e-mail: koyama@tmtv.ne.jp) N. Kurokawa: Department of Mathematics, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo, 152-8551, Japan (e-mail: kurokawa@math.titech.ac.jp)