Affiliation:
1. Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Naugarduko Str. 24, LT-03225 Vilnius, Lithuania
2. Institute of Regional Development, Šiauliai Academy, Vilnius University, Vytauto Str. 84, LT-76352 Šiauliai, Lithuania
Abstract
Suppose that Q is a positive defined n×n matrix, and Q[x̲]=x̲TQx̲ with x̲∈Zn. 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 it has a meromorphic continuation to the whole complex plane. Let n⩾4 be even, while φ(t) is an increasing differentiable function with a continuous monotonic bounded derivative φ′(t) such that φ(2t)(φ′(t))−1≪t, and the sequence {aφ(k)} is uniformly distributed modulo 1. In the paper, it is obtained that 1N#N⩽k⩽2N:ζ(σ+iφ(k);Q)∈A, A∈B(C), for σ>n−12, converges weakly to an explicitly given probability measure on (C,B(C)) as N→∞.
Funder
Research Council of Lithuania
Subject
General Mathematics,Engineering (miscellaneous),Computer Science (miscellaneous)
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