A generalization of the Chowla-Selberg formula.

Abstract

0. Introduction In this paper we prove a formula which generalizes the famous Chowla-Selberg formula to the cases of arbitrary orders of an imaginary quadratic field. It was conjectured by M. Kaneko and has recently been proved also by himself analytically([8]). We recall first the original Chowla-Selberg formula ([14], or Chap. IX of [15]). In what follows we regard a number field as a subfield of the complex number field C. Let K be an imaginary quadratic field, OK its integer ring, h the order of the ideal class group Pic OK , D the absolute value of the discriminant of K, w the order of the unit group of OK , and χ : (Z/DZ) −→ {0,±1} the Kronecker symbol with respect to K over Q. For each lattice L in C, ∆(L) denotes the discriminant of L. If a is a proper ideal of an order of an imaginary quadratic field, F (a) := ∆(a)∆(a−1) depends only on the ideal class of a. Then the Chowla-Selberg formula is