Analytic continuation of Eisenstein series

Abstract:

The classical Eisenstein series are essentially of the form $\Sigma ’_{m, n} \left ( (m + r_1) z + n + r_2\right )^{-s}$, $m$, $n$ ranging over integer values, $\operatorname {Im} z > 0$, $r_1$, $r_2$ rational and $s$ an integer $> 2$. In this paper we show that if $s$ is taken to be complex the series, with ${r_1},{r_2}$ any real numbers, defines an analytic function of $(z,s)$ for $\operatorname {Im} z > 0$, $\operatorname {Re} s > 2$. Furthermore this function has an analytic continuation over the entire $s$ plane, exhibted explicitly by a convergent Fourier expansion. A formula for the transformation of the function when $z$ is subjected to a modular transformation is obtained and the special case of $s$ an integer is studied in detail.