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

Suppose that Q is a positive defined n × n matrix, and Q [ x ̲ ] = x ̲ T Q x ̲ with x ̲ ∈ Z n . The Epstein zeta-function ζ (s ; Q) , s = σ + i t , is defined, for σ > n 2 , by the series ζ (s ; Q) = ∑ x ̲ ∈ Z n ∖ { 0 ̲ } (Q [ x ̲ ]) − 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 (2 t) (φ ′ (t)) − 1 ≪ t , and the sequence { a φ (k) } is uniformly distributed modulo 1. In the paper, it is obtained that 1 N # N ⩽ k ⩽ 2 N : ζ (σ + i φ (k) ; Q) ∈ A , A ∈ B (C) , for σ > n − 1 2 , converges weakly to an explicitly given probability measure on (C , B (C)) as N → ∞ . [ABSTRACT FROM AUTHOR]