Extension of a Bohr-Jessen type theorem for the Epstein zeta-function in short intervals
Let Q be a positive definite n × n matrix, n 2N, n 4. The Epstein zeta-function ζ(s; Q), defined for Re s > n2, is given by ζ(s; Q) = _x Z^n {0 (x^T Qx )^-s, and has a meromorphic continuation to the whole complex plane. Let T^27/82 H T^1/2 . In this paper, we prove a limit theorem on weak convergence for 1H meas{t [T, T+H]: ζ(σ + it; Q) A }, A B(C), as T→∞, where B(C) is the Borel σ-algebra on C. The limit measure is explicitly given. The result extends a known theorem obtained for the interval [0, T]. , number=2, journal=Mathematical Modelling and Analysis, author=Balčiūnas, Aidas and Garbaliauskienė, Virginija and Macaitienė, Renata, year=2026, month=Apr., pages=288–302