98-587 Chang C.-H., Mayer D.H.

The period function of the nonholomorphic Eisenstein series for PSL(2,Z) (16K, LATeX) Sep 3, 98

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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.

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