On a Class of Epstein Zeta Functions

Abstract

X.-J.~Li gave in~[4] a criterion for the Riemann hypothesis in terms of the positivity of a set of coefficients. In this paper, we investigate exactly how the Li criterion for the Riemann hypothesis fails for a class of Epstein zeta functions. This enables to derive some interesting consequences regarding $c_K=\frac{h_K\log d_K}{\sqrt{d_K}}$ of a quadratic imaginary field $K$ of absolute discriminant $d_K$ and class number $h_K$. Similar results are stated for the period ratios of elliptic curves with complex multiplication.