Discrete Limit Bohr–Jessen Type Theorem for the Epstein Zeta-Function in Short Intervals

We prove a probabilistic limit theorem for the Epstein zeta-function ζ(s;Q) in the interval [N,N+M] as N→∞, using discrete shifts ζ(σ+ikh;Q), where h>0 and σ>n−12 are fixed. Here, Q is a positive-definite n×n matrix, and the interval length M satisfies h−1(Nh)27/82⩽M⩽h−1(Nh)1/2. The limit measure is given explicitly. This theorem is the first result in short intervals for ζ(s;Q). The obtained theorem improves the known results established for the interval of length N. Since the considered probability measures are defined in terms of frequency, theorems in short intervals have a certain advantage in the detection of ζ(σ+ikh;Q) with a given property, as well as in the characterization of the asymptotic behaviour of ζ(s;Q) in general.