CHARACTER ANALOGUES OF DEDEKIND SUMS AND TRANSFORMATIONS OF ANALYTIC EISENSTEIN SERIES

In this paper we give a transformation formula for an ana-lytic generalized Eisenstein series. In the same way that the ordinary Dedekind sums arise in the transformation formula for the logarithm of the Dedekind eta-function, similar sums arise in the formula for the following generalized Eisenstein series. 1. Introduction. We put e(z) = e2piiz and let r: = (r1, r2) and h: = (h1, h2). Throughout this paper χ denotes a primitive character of modulus k. We extend the definition of χ by setting χ(x) = 0, if x 6 ∈ Z. For σ = Re(s)> 2 and Im(z)> 0, defin