MPEJ Volume 4, No.6, 8pp
Received: Sep 11, 1998, Accepted: Oct 23, 1998
Cheng-Hung Chang, Dieter Mayer
The period function of the nonholomorphic Eisenstein series for PSL(2,Z)
ABSTRACT: We calculate the period function of Lewis of the automorphic
Eisenstein series $E(s,w)=\frac{1}{2}v^s\,\sum_{n,m\neq (0,0)}(mw+n)^{-2s}$
for the modular group $PSL(2,Z)$. This function turns out to be the
function $B(\frac{1}{2},s+\frac{1}{2})\psi_s(z)$, where $B(x,y)$ denotes
the beta function and $\psi_s$ a function introduced some time ago
by Zagier and given for $\Re s>1$ by the series
$\psi_s(z)=\sum_{n,m\geq 1}(mz+n)^{-2s}+\frac{1}{2}\zeta(2s)\,(1+z^{-2s})$.
The analytic extension of $\psi_s$ to negative integers $s$ gives
just the odd part of the period functions in the Eichler, Shimura, Manin
theory for the holomorphic Eisenstein forms of weight $-2s+2$. We find
this way an interesting connection between holomorphic and nonholomorphic
Eisenstein series on the level of their respective period functions.