EVEN POWERS OF DIVISORS AND ELLIPTIC ZETA VALUES

GIOVANNI FELDER^{1} AND ALEXANDER VARCHENKO^{2}

Abstract. We introduce and study elliptic zeta values, a two-parameter deformation of the values of Riemann’s zeta function at positive integers. They are essentially Taylor coefficients of the logarithm of the elliptic gamma function, and share the SL(3, ℤ) modular properties of this function. Elliptic zeta values at even integers are related to Eisenstein series and thus to sums of odd powers of divisors. The elliptic zeta values at odd integers can be expressed in terms of generating series of sums of even powers of divisors.

Dedicated to Igor Frenkel on the occasion of his 50^{th} birthday

1. Introduction

The generating function of the sum of kth powers of divisors of positive integers

∑_{n=1}^{∞} σ_{k}(n)q^{n}, σ_{k}(n) = ∑_{d | n} d^{k},

converges in |q| < 1. If k is odd, it has interesting transformation properties under the modular group SL(2, ℤ). Indeed, it is closely related to the Eisenstein series G_{k} (see e. g. [2]): let

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

Then

G_{k}(τ) = 2ζ(2k) + 2D_{2k−1}(q), q = e^{2πiτ}, k = 1, 2, 3, …,

where ζ(s) = ∑_{n=1}^{∞} n^{−s} is the Riemann zeta function. For k ≥ 2, these functions are modular forms of weight 2k:

G_{k} (\frac{aτ + b}{cτ + d}) = (cτ + d)^{2k}G_{k}(τ), ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL(2, ℤ),

as can be seen from the representation G_{k}(τ) = ∑_{(n,m)≠(0,0)}(n + mτ)^{−2k}. It can easily be deduced from this that

lim_{τ→0} τ^{2k}D_{2k−1}(q) = ζ(2k).

Date: May 2002.

1Supported in part by the Swiss National Science Foundation. Research at MSRI is supported in part by NSF grant DMS-9810361.

2Supported in part by NSF grant DMS-9801582.

In other words, we may regard τ^{k+1}D_{k}(q) as a one-parameter deformation of the zeta value ζ(k + 1), and this is true also if k is even. For even k, however, D_{k}(q) does not have obvious modular properties. The purpose of this note is to show that D_{k}(q) can be embedded into a two-parameter deformation of ζ(k + 1), the elliptic zeta value at k + 1, which obeys identities of modular type. These identities are essentially equivalent to modularity in the even case but are of a different nature in the odd case. They are related to (and a consequence of) the SL(3, ℤ) properties of the elliptic gamma function [1].

Acknoledgement. The first author wishes to thank Daan Krammer for a useful discussion.

2. Differences of modular forms

Let H be the upper half-plane Im τ > 0.

Proposition 2.1. Let k be a positive integer. Suppose Z(τ, σ) is a holomorphic function on H × H admitting an expansion

Z(τ, σ) = ∑_{n,m=0}^{∞} a_{n,m}q^{n}r^{m}, q = e^{2πiτ}, r = e^{2πiσ},

and obeying Z(τ, σ) = −Z(σ, τ). Then the following statements are equivalent:

(i) Z(τ, σ) = G(τ) − G(σ) for some modular form G of weight 2k.

(ii) Z obeys the three-term relations

Z(τ, σ) = Z(τ, τ + σ) + Z(τ + σ, σ),
Z(τ, σ) = τ^{−2k}Z (−\frac{1}{τ}, \frac{σ}{τ} ) + σ^{−2k}Z (−\frac{τ}{σ} , −\frac{1}{σ} ) ,

for all σ, τ ∈ H such that σ/τ ∈ H.

Proof: It is easy to check that if G is a modular form then G(τ) − G(σ) obeys the threeterm relations. Conversely, let us extend a_{n,m} to all integers n, m by setting a_{n,m} = 0 if n or m is negative. The first three-term relation implies that a_{n−m,m} + a_{n,m−n} = a_{n,m}. It follows that a_{n,m} = a_{n−m,m} if n > m and a_{n,m} = a_{n,m−n} if m > n. By the Euclidean algorithm we see that for n, m > 0, a_{n,m} = a_{N,N}, where N = (n, m) is the greatest common divisor. But a_{N,N} = 0 since Z is odd under interchange of q and r. Thus only a_{n,0}, a_{0,m} may be non-zero and Z(τ, σ) = G(τ) − G(σ) for some function G admitting an expansion in powers of q. In particular, G(τ + 1) = G(τ). The second three-term relation then implies that G(−1/τ) = τ^{2k}G(τ) possibly after redefining G by adding a constant. Thus G is a modular form of weight 2k. □

3. Elliptic zeta values

We now define two-parameter deformations of the values of the zeta function at positive integers, which we call elliptic zeta values:

(1) Z_{n}(τ, σ) = −\frac{(2πi)^{n}}{(n − 1)!} ∑_{j=1}^{∞} j^{n−1} \frac{q^{j} − (−1)^{n}r^{j}}{(1 − q^{j})(1 − r^{j})} , q = e^{2πiτ}, r = e^{2πiσ}, n ∈ ℤ_{≥1}.

Clearly, Z_{n}(τ, σ) = −(−1)^{n}Z_{n}(σ, τ). The relation to the functions D_{k} is obtained by expanding the elliptic zeta values in a power series in q and r. The result is the following.

If n = 2k is even, then

Z_{2k}(τ, σ) = D_{2k−1}(r) − D_{2k−1}(q).

If n = 2k + 1 is odd, then

Z_{2k+1}(τ, σ) = D_{2k}(q) + D_{2k}(r) + 2 ∑_{(a,b)=1} D_{2k}(q^{a}r^{b}).

The sum is over pairs of relatively prime pairs of positive integers a, b.

Theorem 3.1.

(i) Let n ≥ 4. Then Z_{n} obeys the three-term relations

Z_{n}(τ, σ) = Z_{n}(τ, τ + σ) + Z_{n}(τ + σ, σ),
Z_{n}(τ, σ) = τ^{−n}Z_{n} (−\frac{1}{τ}, \frac{σ}{τ} ) + (−σ)^{−n}Z_{n} (−\frac{τ}{σ} , −\frac{1}{σ} ) ,

for all σ, τ ∈ H such that σ/τ ∈ H. (ii) lim_{σ→0} lim_{τ→i∞} σ^{n}Z_{n}(τ, σ) = ζ(n) if n ≥ 2. (iii) For n = 1, lim_{σ→0} lim_{τ→i∞}(σZ_{1}(τ, σ) + ln(−2πiσ)) is the Euler constant γ =

0.577 … (the logarithm is real if σ is imaginary).

In particular, by taking the limit of Z_{n} as τ → i∞ we obtain

lim_{σ→0}(σD_{0}(e^{2πiσ}) + ln(−2πiσ)) = γ,
lim_{σ→0} σ^{n+1}D_{n}(e^{2πiσ}) = ζ(n + 1),  n = 1, 2, …

We prove Theorem 3.1 in the next section.

4. The elliptic gamma function

Theorem 3.1 follows from the results of [1] where the modular properties of Ruijsenaars’ elliptic gamma function were discovered. In the normalization of [1], the elliptic gamma function is defined by the double infinite product

Γ(z, τ, σ) = ∏_{j,k=0}^{∞} \frac{1 − q^{j+1}r^{k+1}e^{−2πiz}}{1 − q^{j}r^{k}e^{2πiz}} z ∈ ℂ, τ, σ ∈ H.

It obeys the functional relation Γ(z + σ, τ, σ) = θ_{0}(z, τ)Γ(z, τ, σ), which is an elliptic version of Euler’s functional equation for the gamma function. Here θ_{0} denotes the theta function

θ_{0}(z, τ) = ∏_{j=0}^{∞} (1 − q^{j+1}e^{−2πiz})(1 − q^{j}e^{2πiz}) z ∈ ℂ, τ ∈ H.

The Euler gamma function Γ(z) is recovered in the limit

(2)  Γ(z) = lim_{σ→0} lim_{τ→i∞} θ_{0}(σ, τ)^{1−z} \frac{Γ(σz, τ, σ)}{Γ(σ, τ, σ)} .

Theorem 4.1. ([1], p. 54) Suppose τ, σ, σ/τ ∈ H. Then

Γ(z, τ, σ) = Γ(z + τ, τ, σ + τ)Γ(z, τ + σ, σ),
Γ(z/τ, −1/τ, σ/τ) = e^{iπQ(z;τ,σ)}Γ((z − τ)/σ, −τ/σ, −1/σ)Γ(z, τ, σ),

for some polynomial Q(z, τ, σ) of degree three in z with coefficients in ℚ(τ, σ).

It follows from the Weierstrass product representation Γ(z + 1) = e^{−γz} ∏_{j=1}^{∞}(1 + z/j)^{−1}e^{jz} that the values of ζ at positive integers are essentially Taylor coefficients at 1 of the logarithm of the Euler gamma function:

(3)  ln Γ(z + 1) = −γz + ∑_{j=2}^{∞} \frac{ζ(j)}{j}(−z)^{j}.

Here γ = lim_{n→∞}(∑_{j=1}^{n−1} 1/j − ln n) is the Euler constant. The elliptic analog of this formula involves the elliptic zeta values:

(4)  ln \frac{Γ(z + σ, τ, σ)}{Γ(σ, τ, σ)} = ∑_{j=1}^{∞} \frac{Z_{j}(τ, σ)}{j} (−z)^{j}.

This formula, with Z_{j} defined by eq. (1), is an easy consequence of the summation formula for ln Γ(z, τ, σ) ([1] p. 51):

ln Γ(z, τ, σ) = −\frac{i}{2} ∑_{j=1}^{∞} \frac{sin(πj(2z − τ − σ))}{j sin(πjτ) sin(πjσ)}.

Proof of Theorem 3.1: The first claim of the theorem is proven by taking the logarithm of the identities of Theorem 4.1 and expanding them in powers of z − σ.

Taking the limit of (4), by using (2) to compare it with (3) implies (ii).

In the same way we can deduce (iii). However, if n = 1, we have to take into account the factor θ_{0}^{1−z} in (2): we obtain γ = lim_{σ→0} lim_{τ→∞}(σZ_{1}(τ, σ) + ln θ_{0}(σ, τ)). Since ln θ_{0}(σ, τ) = ln(−2πiσ) plus terms that vanish in the limit, the proof is complete. □

By using the explicit formula for the polynomial Q, we also obtain the exceptional relations for n = 1, 2, 3:

Theorem 4.2. Let 1 ≤ n ≤ 3 and suppose τ, σ, σ/τ ∈ H. Then

Z_{n}(τ, σ) = Z_{n}(τ, τ + σ) + Z_{n}(τ + σ, σ),
Z_{n}(τ, σ) = τ^{−n}Z_{n} (−\frac{1}{τ}, \frac{σ}{τ} ) + (−σ)^{−n}Z_{n} (−\frac{τ}{σ} , −\frac{1}{σ} ) + iπa_{n},

where

a_{1} = −\frac{1}{2} + \frac{1}{2τ} − \frac{1}{2σ} + \frac{σ}{6τ} + \frac{τ}{6σ} + \frac{1}{6τσ} ,
a_{2} = −\frac{1}{τ} + \frac{1}{σ} − \frac{1}{τσ} ,
a_{3} = \frac{1}{τσ} .

References

[1] G. Felder and A. Varchenko, The elliptic gamma function and SL(3, ℤ) ⋉ ℤ^{3}, Adv. Math. 156 (2000), 44–76

[2] J.-P. Serre, A course in arithmetic, Springer 1973

G. F.: MSRI, 1000 Centennial Drive, Berkeley, CA 94720, USA and Departement of Mathematics, ETH-Zentrum, 8092 Zürich, Switzerland

E-mail address: felder@math.ethz.ch

A. V.: Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA

E-mail address: anv@email.unc.edu