A Generalized Bohr–Jessen Type Theorem for the Epstein Zeta-Function

Abstract

Let Q be a positive defined n×n matrix and Q[x̲]=x̲TQx̲. The Epstein zeta-function ζ(s;Q), s=σ+it, is defined, for σ>n2, by the series ζ(s;Q)=∑x̲∈Zn\{0̲}(Q[x̲])−s, and is meromorphically continued on the whole complex plane. Suppose that n⩾4 is even and φ(t) is a differentiable function with a monotonic derivative. In the paper, it is proved that 1Tmeast∈[0,T]:ζ(σ+iφ(t);Q)∈A, A∈B(C), converges weakly to an explicitly given probability measure on (C,B(C)) as T→∞.