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\title[{}]{Even powers of divisors and elliptic zeta values}
\author[{}]
{Giovanni Felder${}^{1}$ 
\and Alexander Varchenko${}^{2}$}
\thanks{${}^1$Supported in part by the Swiss National Science Foundation.
Research at MSRI is supported in part by NSF grant DMS-9810361} 
\thanks{${}^2$Supported in part by NSF grant  DMS-9801582}
\address{G. F.: MSRI, 1000 Centennial Drive, Berkeley, CA 94720, USA
and
Departement of Mathematics,
  ETH-Zentrum, 8092 Z\"urich, Switzerland}
\email{felder@math.ethz.ch}
\address{A. V.: Department of Mathematics, University of
  North Carolina at Chapel Hill, Chapel Hill, NC
  27599-3250, USA} 
\email{anv@email.unc.edu}
\date{May 2002}
\begin{document}
\begin{abstract}
We introduce and study {\em 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($3,\mathbb{Z}$) 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.
\end{abstract}
\maketitle

\newcommand{\Matrix}[4]{\left(\begin{array}{cc}#1&#2\\ #3&#4\end{array}\right)}
\centerline{\it Dedicated to Igor Frenkel on the occasion of his
$50^{\text{th}}$ birthday}

\section{Introduction}
The generating function of the sum of $k$th powers of divisors of
positive integers
\[
\sum_{n=1}^\infty\sigma_k(n)q^n,\qquad
\sigma_k(n)=\sum_{d\,|\,n}d^k,
\]
converges in $|q|<1$. If $k$ is odd,
it has interesting transformation properties 
under the modular
group $\mathrm{SL}(2,\Z)$. 
Indeed, it is closely related to the Eisenstein series $G_k$
(see e.~g.~\cite{Serre}): let
\[
D_k(q)=\frac{(-2\pi i)^{k+1}}{k!}\sum_{n=1}^\infty\sigma_k(n)q^n.
\]
Then
\[
G_k(\tau)=2\zeta(2k)+2D_{2k-1}(q),
\qquad q=e^{2\pi i\tau},\qquad k=1,2,3,\dots,
\]
where $\zeta(s)=\sum_{n=1}^\infty n^{-s}$ is the
Riemann zeta function. For $k\geq2$, 
these functions are modular forms of weight $2k$:
\[
G_k\left(\frac{a\tau+b}{c\tau+d}\right)=(c\tau+d)^{2k}G_k(\tau),
\qquad\Matrix abcd\in \mathrm{SL}(2,\Z),
\]
as can be seen from the representation
$G_k(\tau)=\sum_{(n,m)\neq(0,0)}(n+m\tau)^{-2k}$. 
It can easily be deduced from this that
\[
\lim_{\tau\to 0}\tau^{2k}D_{2k-1}(q)
=\zeta(2k).
\]
In other words, we may regard $\tau^{k+1}D_k(q)$
as a one-parameter deformation of the zeta value $\zeta(k+1)$, 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 $\zeta(k+1)$,
the {\em 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 $\mathrm{SL}(3,\Z)$ properties 
of the elliptic gamma function \cite{FV}.
\subsection*{Acknoledgement} The first author wishes to thank Daan Krammer for
a useful discussion.

\section{Differences of modular forms}
Let $H$ be the upper half-plane 
$\mathrm{Im}\,\tau>0$.

\begin{proposition} Let $k$ be a positive integer.
Suppose $Z(\tau,\sigma)$ is a holomorphic
function on $H\times H$ admitting an expansion
\[
Z(\tau,\sigma)=\sum_{n,m=0}^\infty
a_{n,m}q^nr^m,\qquad
q=e^{2\pi i\tau},\quad r=e^{2\pi i\sigma},
\]
and obeying $Z(\tau,\sigma)=-Z(\sigma,\tau)$.
Then the following statements are equivalent:
\begin{enumerate} 
\item[(i)] $Z(\tau,\sigma)=G(\tau)-G(\sigma)$ for
some modular form $G$ of weight $2k$.
\item[(ii)] $Z$ obeys the three-term relations
\begin{eqnarray*}
Z(\tau,\sigma)&=&Z(\tau,\tau+\sigma)+Z(\tau+\sigma,\sigma),\\
Z(\tau,\sigma)&=&
\tau^{-2k}Z\left(-\frac1\tau,\frac\sigma\tau\right)
+\sigma^{-2k}Z\left(-\frac\tau\sigma,-\frac1\sigma\right),
\end{eqnarray*}
for all $\sigma,\tau\in H$ such that $\sigma/\tau\in H$.
\end{enumerate}
\end{proposition}

\noindent{\it Proof:}
It is easy to check that if $G$ is a modular form then
$G(\tau)-G(\sigma)$ obeys the three-term 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 
$Z(\tau,\sigma)=G(\tau)-G(\sigma)$ for some function
$G$ admitting an  expansion in powers of $q$.
In particular, $G(\tau+1)=G(\tau)$. 
The second three-term relation
then implies that $G(-1/\tau)=\tau^{2k}G(\tau)$ possibly after
 redefining $G$ by adding a constant.
Thus $G$ is a modular form of weight $2k$.
\hfill $\square$



\section{Elliptic zeta values}
We now define two-parameter deformations of the
values of the zeta function at positive integers,
which we call {\em elliptic zeta values}:
\begin{equation}\label{e-EZV}
Z_n(\tau,\sigma)=
-\frac{(2\pi i)^n}{(n-1)!}
\sum_{j=1}^\infty
j^{n-1}
\frac{q^j-(-1)^nr^j}{(1-q^j)(1-r^j)}\,,\quad
q=e^{2\pi i\tau},\quad r=e^{2\pi i\sigma}, \quad n\in\Z_{\geq1}.
\end{equation}
Clearly, $Z_n(\tau,\sigma)=-(-1)^nZ_n(\sigma,\tau)$.
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
\[
Z_{2k}(\tau,\sigma)=D_{2k-1}(r)-D_{2k-1}(q).
\]
 If $n=2k+1$ is odd, then 
\[
Z_{2k+1}(\tau,\sigma)=
D_{2k}(q)+D_{2k}(r)+2\sum_{(a,b)=1}D_{2k}(q^ar^b).
\]
The sum is over pairs of relatively prime pairs of
positive  integers $a,b$.

\begin{thm}\label{t-1}\ 
\begin{enumerate}
\item[(i)] Let $n\geq4$. Then
$Z_n$ obeys the three-term relations
\begin{eqnarray*}
Z_n(\tau,\sigma)&=&Z_n(\tau,\tau+\sigma)+Z_n(\tau+\sigma,\sigma),\\
Z_n(\tau,\sigma)&=&
\tau^{-n}Z_n\left(-\frac1\tau,\frac\sigma\tau\right)
+(-\sigma)^{-n}Z_n\left(-\frac\tau\sigma,-\frac1\sigma\right),
\end{eqnarray*}
for all $\sigma,\tau\in H$ such that $\sigma/\tau\in H$.
\item[(ii)] $\lim_{\sigma\to 0}\lim_{\tau\to i\infty}
\sigma^nZ_n(\tau,\sigma) =\zeta(n)$ if $n\geq2$.
\item[(iii)]
For $n=1$,  $\lim_{\sigma\to 0}\lim_{\tau\to i\infty}
(\sigma Z_1(\tau,\sigma)+\ln(-2\pi i\sigma))$ is
the Euler constant $\gamma=0.577\dots$ (the logarithm is
real if $\sigma$ is imaginary).
\end{enumerate}
\end{thm}

In particular, by taking the limit of $Z_n$ as $\tau\to i\infty$ we 
obtain
\begin{eqnarray*}
\lim_{\sigma\to0}(\sigma D_0(e^{2\pi i\sigma})+\ln(-2\pi i\sigma))
&=&
\gamma, \\
\lim_{\sigma\to0}\sigma^{n+1}D_n(e^{2\pi i\sigma})&=&
\zeta(n+1), \qquad n=1,2,\dots
\end{eqnarray*}
We prove Theorem \ref{t-1} in the next section.
\section{The elliptic gamma function}
Theorem \ref{t-1} follows from the results of \cite{FV}
where the modular properties of Ruijsenaars' elliptic gamma
function 
were discovered. In the normalization of \cite{FV}, the
elliptic gamma function is defined by the double
infinite product
\[
\Gamma(z,\tau,\sigma)=\prod_{j,k=0}^\infty
\frac
{1-q^{j+1}r^{k+1}e^{-2\pi iz}}
{1-q^{j}  r^{k}  e^{ 2\pi iz}}\,\qquad z\in\C,\tau,\sigma\in H.
\]
It obeys the functional relation
$\Gamma(z+\sigma,\tau,\sigma)
=\theta_0(z,\tau)\Gamma(z,\tau,\sigma),
$
which is an elliptic version of Euler's functional equation
for the gamma function. Here $\theta_0$ denotes the theta
function
\[
\theta_0(z,\tau)=
\prod_{j=0}^\infty
{(1-q^{j+1}e^{-2\pi iz})}
{(1-q^{j}e^{ 2\pi iz})}\,\qquad z\in\C,\tau\in H.
\]
The Euler gamma function $\Gamma(z)$ is recovered in the limit
\begin{equation}\label{e-SCL}
\Gamma(z)=\lim_{\sigma\to 0}
\lim_{\tau\to i\infty}\theta_0(\sigma,\tau)^{1-z}
\frac
{\Gamma(\sigma z,\tau,\sigma)}
{\Gamma(\sigma  ,\tau,\sigma)}\,.
\end{equation}
\begin{thm}\label{t-2} $($\cite{FV}, p.~54$)$ Suppose 
$\tau,\sigma,\sigma/\tau\in H$. Then
\begin{eqnarray*}
{\Gamma(z,\tau,\sigma)}
&=&
{\Gamma(z+\tau,\tau,\sigma+\tau)}\Gamma(z,\tau+\sigma,\sigma),
\\
\Gamma(
 z/\tau,-1/\tau,\sigma/\tau)&=&
e^{i\pi Q(z;\tau,\sigma)}\label{e-u}
{\Gamma(
{({z-\tau})/\sigma,-\tau/\sigma,-1/\sigma})}
{\Gamma(z,\tau,\sigma)},
\end{eqnarray*}
for some polynomial $Q(z,\tau,\sigma)$ of degree three
in $z$ with coefficients in $\Q(\tau,\sigma)$.
\end{thm}

It follows from the Weierstrass product representation
$\Gamma(z+1)=e^{-\gamma z}
\prod_{j=1}^\infty(1+z/j)^{-1}e^{jz}$
that the values of $\zeta$ at positive integers are
essentially Taylor coefficients at $1$ of the logarithm
of the Euler gamma function:
\begin{equation}\label{e-logG}
\ln\,\Gamma(z+1)=-\gamma z+\sum_{j=2}^\infty\frac{\zeta(j)}j
(-z)^j.
\end{equation}
Here $\gamma=\lim_{n\to\infty}(\sum_{j=1}^{n-1}1/j-\ln\,n)$ is
the Euler constant.
The elliptic analog of this formula involves the elliptic
zeta values:
\begin{equation}\label{e-logEG}
\ln
\frac
{\Gamma(z+\sigma,\tau,\sigma)}
{\Gamma(\sigma,\tau,\sigma)}=
\sum_{j=1}^\infty \frac{Z_j(\tau,\sigma)}{j}\, (-z)^j.
\end{equation}
This formula, with $Z_j$ defined by eq.~\eqref{e-EZV},
is an easy consequence of the summation formula for
$\ln\,\Gamma(z,\tau,\sigma)$ (\cite{FV} p.~51):
\[
\ln\,\Gamma(z,\tau,\sigma)=
-\frac i2\sum_{j=1}^\infty
\frac{\sin(\pi j(2z-\tau-\sigma))}
{j\sin(\pi j\tau)\sin(\pi j\sigma )}.
\]
{\it Proof of Theorem \ref{t-1}:}\/
The first claim of the theorem 
is proven by taking the
logarithm of the identities of Theorem \ref{t-2} and
expanding them in powers of $z-\sigma$.

 Taking the limit of \eqref{e-logEG}, by using
\eqref{e-SCL} to compare it with \eqref{e-logG} implies
(ii).

In the same way we can deduce (iii).
However, if $n=1$, we have to take into
account the factor $\theta_0^{1-z}$ in
\eqref{e-SCL}: we obtain $\gamma=\lim_{\sigma\to 0}
\lim_{\tau\to\infty}(\sigma Z_1(\tau,\sigma)+
\ln\theta_0(\sigma,\tau))$. 
Since $\ln\theta_0(\sigma,\tau)=\ln(-2\pi i\sigma)$
plus terms that vanish in the limit, the proof is complete.
\hfill$\square$
\medskip

By using the explicit formula for the polynomial $Q$,
we also obtain the exceptional  relations for $n=1,2,3$:
\begin{thm}\label{t-3}
Let $1\leq n\leq3$ and suppose $\tau,\sigma,\sigma/\tau\in H$. 
Then
\begin{eqnarray*}
Z_n(\tau,\sigma)&=&Z_n(\tau,\tau+\sigma)+Z_n(\tau+\sigma,\sigma),\\
Z_n(\tau,\sigma)&=&
\tau^{-n}Z_n\left(-\frac1\tau,\frac\sigma\tau\right)
+(-\sigma)^{-n}Z_n\left(-\frac\tau\sigma,-\frac1\sigma\right)
+i\pi a_n,
\end{eqnarray*}
where
\begin{eqnarray*}
a_1&=&
-\frac12
+\frac1{2\tau}
-\frac1{2\sigma}
+\frac{\sigma}{6\tau}
+\frac{\tau}{6\sigma}
+\frac1{6\tau\sigma}\,,
\\
a_2&=&-\frac1\tau+\frac1\sigma-\frac1{\tau\sigma}\,,
\\
a_3&=&\frac1{\tau\sigma}\,.
\end{eqnarray*}
\end{thm}

\begin{thebibliography}{CC}
\bibitem{FV}
G. Felder and A. Varchenko,
{\em The elliptic gamma function and $\mathrm{SL}(3,\Z)\semidirect\Z^3$},
Adv.\ Math.\ 156 (2000), 44--76
\bibitem{Serre} J.-P. Serre, {\em A course in arithmetic,}
Springer 1973
\end{thebibliography}

\end{document}