Multiple Eisenstein Series and Multiple Cotangent Functions

Shin-ya Koyama and Nobushige Kurokawa

Abstract. We construct the multiple Eisenstein series and we show a relation to the multiple cotangent function. We calculate a limit value of the multiple Eisenstein series.

Key words: multiple Eisenstein series, multiple cotangent function, multiple polylogarithm

AMS Classification 11M06

1 Introduction

For integers r ≥ 1 and k ≥ r + 2, we define the multiple Eisenstein series as

F_{k}(τ_{1}, ⋯, τ_{r}) = ∑_{n=1}^{∞} \frac{n^{k−1}q_{1}^{n} ⋯ q_{r}^{n}}{(1 − q_{1}^{n}) ⋯ (1 − q_{r}^{n})}

with q_{j} = e^{2πiτ_{j}} for Im(τ_{j}) > 0. We remark that the series converges for (τ_{1}, ⋯, τ_{r}) ∈ (C − R_{≤0})^{r} if Im(τ_{j}) > 0 for at least one j. This function is considered to be the multiple q (quantum) polylogarithm since

F_{k}(τ_{1}, ⋯, τ_{r}) = \frac{1}{(1 − q_{1}) ⋯ (1 − q_{r})} ∑_{n=1}^{∞} \frac{q_{1}^{n} ⋯ q_{r}^{n}}{n^{1−k}[n]_{q_{1}} ⋯ [n]_{q_{r}}} ,

where

[x]_{q} = \frac{1 − q^{x}}{1 − q} .

We recall the one-variable case

F_{k}(τ) = ∑_{n=1}^{∞} \frac{n^{k−1}q^{n}}{1 − q^{n}} = ∑_{n=1}^{∞} σ_{k−1}(n)q^{n}

with

σ_{k−1}(n) = ∑_{d | n} d^{k−1}

and

q = e^{2πiτ}

occurs frequently in the form

E_{k}(τ) = \frac{ζ(1 − k)}{2} + ∑_{n=1}^{∞} σ_{k−1}(n)q^{n}

with the constant term ζ(1 − k)/2.

In a previous paper [K2] we proved that

lim_{{τ→1 \atop Im(τ)>0}} (E_{k} (−\frac{1}{τ}) − τ^{k}E_{k}(τ)) = \frac{(−1)^{k}B_{k−1}}{2πi}

for all positive integers k. This is checked for even integers k by the modularity of E_{k}(τ) for SL_{2}(Z), but the above limit value seems to be curious especially for odd integers k. We remark that the above result is written also as

lim_{{τ→1 \atop Im(τ)>0}} (F_{k} (−\frac{1}{τ}) − τ^{k}F_{k}(τ)) = \frac{(−1)^{k}B_{k−1}}{2πi} .

In this paper we generalize this result to the case of several variables.

To describe our results we introduce multiple sine functions and multiple cotangent functions. The multiple sine function S_{r}(x, (ω_{1}, ⋯, ω_{r})) is constructed as

S_{r}(x, (ω_{1}, ⋯, ω_{r}))
= ∏ ∐_{n_{1},··· ,n_{r}≥0}(n_{1}ω_{1} + ⋯ + n_{r}ω_{r} + x) ( ∏ ∐_{m_{1},··· ,m_{r}≥1}(m_{1}ω_{1} + ⋯ + m_{r}ω_{r} − x))^{(−1)^{r−1}} ,

where ∏ ∐ denotes the regularized product of Deninger [D]:

∏ ∐_{λ}λ = exp (−\frac{d}{ds} ∑_{λ} λ^{−s}|_{s=0}) .

We refer to [K1] [KK] for detailes of the theory of multiple sine functions; see the survey of Manin [M]. The multiple cotangent function Cot_{r}(x, (ω_{1}, ⋯, ω_{r})) is defined as the logarithmic derivative of the multiple sine function:

Cot_{r}(x, (ω_{1}, ⋯, ω_{r})) = \frac{S′_{r}(x, (ω_{1}, ⋯, ω_{r}))}{S_{r}(x, (ω_{1}, ⋯, ω_{r}))}.

Our first result relates F_{k}(τ_{1}, ⋯, τ_{r}) to Cot_{r+1}^{(k−1)}(τ_{1} + ⋯ + τ_{r}, (τ_{1}, ⋯, τ_{r}, 1)).

Theorem 1 Let k ≥ r + 2 and 0 < arg(τ_{1}) < ⋯ < arg(τ_{r}) < π. Then

Cot_{r+1}^{(k−1)}(τ_{1} + ⋯ + τ_{r}, (τ_{1}, ⋯, τ_{r}, 1))

= −(2πi)^{k} (F_{k}(τ_{1}, ⋯, τ_{r}) − \frac{1}{τ_{1}^{k}} F_{k} (−\frac{1}{τ_{1}} , \frac{τ_{2}}{τ_{1}} , ⋯, \frac{τ_{r}}{τ_{1}} ) − \frac{1}{τ_{2}^{k}} F_{k} (\frac{τ_{1}}{τ_{2}} , −\frac{1}{τ_{2}} , \frac{τ_{3}}{τ_{2}} , ⋯, \frac{τ_{r}}{τ_{2}} )

− ⋯ − \frac{1}{τ_{r}^{k}} F_{k} (\frac{τ_{1}}{τ_{r}} , ⋯, \frac{τ_{r−1}}{τ_{r}} , −\frac{1}{τ_{r}} )) .

Examples (1)

Cot_{2}^{(k−1)}(τ, (τ, 1)) = −(2πi)^{k} (F_{k}(τ) − \frac{1}{τ^{k}} F_{k} (−\frac{1}{τ})) .

(2)

Cot_{3}^{(k−1)}(τ + σ, (τ, σ, 1)) = −(2πi)^{k} (F_{k}(τ, σ) − \frac{1}{τ^{k}} F_{k} (−\frac{1}{τ}, \frac{σ}{τ} ) − \frac{1}{σ^{k}} F_{k} (\frac{τ}{σ} , −\frac{1}{σ} )) .

Theorem 2 Let k ≥ r + 2. Then

lim_{τ_{1},··· ,τ_{r}→1} (F_{k}(τ_{1}, ⋯, τ_{r}) − \frac{1}{τ_{1}^{k}} F_{k} (−\frac{1}{τ_{1}} , \frac{τ_{2}}{τ_{1}} , ⋯, \frac{τ_{r}}{τ_{1}} ) − ⋯ − \frac{1}{τ_{r}^{k}} F_{k} (\frac{τ_{1}}{τ_{r}} , ⋯, \frac{τ_{r−1}}{τ_{r}} , −\frac{1}{τ_{r}} ))
= \frac{(−1)^{r−1}(k − 1)!}{r!(2πi)^{k}} ∑_{m=0}^{[ \frac{k−r}{2} ]} \frac{(−1)^{m}π^{2m}a(r, k − 2m)}{(2m)!}B_{2m},

where a(r, j) ∈ Z is a Stirling like number defined by

x(x + 1) ⋯ (x + r − 1) = ∑_{j=1}^{n} a(r, j)x^{j}.

Examples (1)

lim_{{τ→1 \atop Im(τ)>0}} (F_{k}(τ) − \frac{1}{τ^{k}} F_{k} (−\frac{1}{τ})) = \frac{(−1)^{k−1}B_{k−1}}{2πi} .

lim {τ,σ→1 \atop 0<arg(τ)<arg(σ)<π} (F_{k}(τ, σ) − \frac{1}{τ^{k}} F_{k} (−\frac{1}{τ}, \frac{σ}{τ} ) − \frac{1}{σ^{k}} F_{k} (\frac{τ}{σ} , −\frac{1}{σ} ))

= \begin{cases}−\frac{B_{k−1}}{4πi} & if k ≥ 1 is odd, \\ \frac{(k − 1)B_{k−2}}{8π^{2}} & if k ≥ 2 is even. \end{cases}

Our functions can be investigated from the viewpoint of the “Stirling modular form” which generalizes a notion used by Barnes [B]. Instead of going into the detailed theory, here we only indicate that “Stirling modular forms” mean some suitable functions of “semilattices” similar to the situation of usual modular forms which are functions of “lattices”. We notice one example which shows that F_{k}(τ_{1}, ⋯, τ_{r}) is a function of the “semi-lattice” Z_{≥1}τ_{1} + ⋯ + Z_{≥1}τ_{r} + Z · 1.

Theorem 3 Let k ≥ r + 2 and Im(τ_{1}), ⋯, Im(τ_{r}) > 0. Then

F_{k}(τ_{1}, ⋯, τ_{r}) = \frac{(−1)^{k}(k − 1)!}{(2πi)^{k}} ∑_{m_{1},··· ,m_{r}≥1} ∑_{n=−∞}^{∞} \frac{1}{(m_{1}τ_{1} + ⋯ + m_{r}τ_{r} + n)^{k}} .

2 Proof of Theorem 1

As proved in [KW] we have

log S_{r}(x, (ω_{1}, ⋯, ω_{r})) = − ∑_{l=1}^{r} ∑_{n=1}^{∞} \frac{e^{\frac{2πinx}{ω_{l}}}}{n ∏_{j≠l} (1 − e^{\frac{2πinω_{j}}{ω_{l}}} )} + Q(x),

where Q(x) is a polynomial of x with deg Q ≤ r. Hence we have

Cot_{r}(x, (ω_{1}, ⋯, ω_{r})) = −2πi ∑_{l=1}^{r} ∑_{n=1}^{∞} \frac{e^{\frac{2πinx}{ω_{l}}}}{ω_{l} ∏_{j≠l} (1 − e^{\frac{2πinω_{j}}{ω_{l}}} )} + Q′(x)

and

Cot_{r}^{(k−1)}(x, (ω_{1}, ⋯, ω_{r})) = −(2πi)^{k} ∑_{l=1}^{r} ∑_{n=1}^{∞} \frac{n^{k−1}e^{\frac{2πinx}{ω_{l}}}}{ω_{l}^{k} ∏_{j≠l} (1 − e^{\frac{2πinω_{j}}{ω_{l}}} )} + Q^{(k)}(x)

for k ≥ 1. In particular

Cot_{r}^{(k−1)}(x, (ω_{1}, ⋯, ω_{r})) = −(2πi)^{k} ∑_{l=1}^{r} ∑_{n=1}^{∞} \frac{n^{k−1}e^{\frac{2πinx}{ω_{l}}}}{ω_{l}^{k} ∏_{j≠l} (1 − e^{\frac{2πinω_{j}}{ω_{l}}} )}

for k ≥ r + 1. Thus, changing r to r + 1 and taking (ω_{1}, ⋯, ω_{r+1}) = (τ_{1}, ⋯, τ_{r}, 1) with x = τ_{1} + ⋯ + τ_{r}, we obtain Theorem 1.

3 Proof of Theorem 2

Theorem 1 shows that the limit value is given by

−\frac{1}{(2πi)^{k}} Cot_{r+1}^{(k−1)}(r, (1, ⋯, 1)).

We calculate it by using the formula

Cot_{r+1}(x, (1, ⋯, 1)) = (−1)^{r}\binom{x − 1}{r}π cot(πx)

proved in [KK]. We look at

Cot_{r+1}(r + x, (1, ⋯, 1)) = (−1)^{r}\binom{x + r − 1}{r}π cot(πx)

and its Taylor expansion around x = 0. Since

\binom{x + r − 1}{r} = \frac{x(x + 1) ⋯ (x + r − 1)}{r!} = \frac{1}{r!} ∑_{j=1}^{r} a(r, j)x^{j}

and

cot(πx) = ∑_{m=0}^{∞} \frac{(−1)^{m}π^{2m−1}B_{2m}}{(2m)!} x^{2m−1}

we have

Cot_{r+1}^{(k−1)}(r, (1, ⋯, 1)) = (k − 1)!\frac{(−1)^{r}π}{r!} ∑_{j=1}^{r} ∑_{m=0}^{∞} \frac{(−1)^{m}π^{2m−1}B_{2m}a(r, j)}{(2m)!}

= (k − 1)!\frac{(−1)^{r}}{r!} ∑_{m=0}^{[ \frac{k−r}{2} ]} \frac{(−1)^{m}π^{2m}a(r, k − 2m)}{(2m)!}B_{2m}.

Thus we have Theorem 2.

We notice that Examples are easily checked by Theorem 1.

4 Proof of Theorem 3

The well-known Lipschitz formula says that

∑_{n=−∞}^{∞}(τ + n)^{−k} = (−1)^{k} \frac{(2πi)^{k}}{(k − 1)!} ∑_{n=1}^{∞} n^{k−1}q^{n}

for Im(τ) > 0 with q = e^{2πiτ}. Especially, for τ = m_{1}τ_{1} + ⋯ + m_{r}τ_{r} we have

∑_{n=−∞}^{∞}(m_{1}τ_{1} + ⋯ + m_{r}τ_{r} + n)^{−k} = (−1)^{k} \frac{(2πi)^{k}}{(k − 1)!} ∑_{n=1}^{∞} n^{k−1}(q_{1}^{m_{1}} ⋯ q_{r}^{m_{r}})^{n}

with q_{j} = e^{2πiτ_{j}}. Hence we obtain

∑_{m_{1},··· ,m_{r}≥1} ∑_{n=−∞}^{∞}(m_{1}τ_{1} + ⋯ + m_{r}τ_{r} + n)^{−k} = (−1)^{k} \frac{(2πi)^{k}}{(k − 1)!} ∑_{n=1}^{∞} \frac{n^{k−1}q_{1}^{n} ⋯ q_{r}^{n}}{(1 − q_{1}^{n}) ⋯ (1 − q_{r}^{n})}.

This gives Theorem 3.

References

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

[D] Deninger, C.: Local L-factors of motives and regularized determinant. Invent. Math. 107 (1992) 135-150.

[K1] Kurokawa, N.: Gamma factors and Plancherel measures. Proc. Japan Acad. 68A (1992) 256-260.

[K2] Kurokawa, N.: Limit values of Eisenstein series and multiple cotangent functions (preprint 2006).

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

[KW] Kurokawa, N., and Wakayama, M.: Absolute tensor products. Int. Math. Res. Not. 2004-5 (2004) 249-260.

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

Department of Mathematics, Tokyo Institute of Technology Oo-okayama, Meguro-ku, Tokyo 152-8551, Japan kurokawa@math.titech.ac.jp