 98587 Chang C.H., Mayer D.H.
 The period function of the nonholomorphic Eisenstein series for PSL(2,Z)
(16K, LATeX)
Sep 3, 98

Abstract ,
Paper (src),
View paper
(auto. generated ps),
Index
of related papers

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 $PSL(2,\zz)$. 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.
 Files:
98587.tex