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

Antanas Laurinčikas and Renata Macaitienė ()
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Antanas Laurinčikas: Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Naugarduko Str. 24, LT-03225 Vilnius, Lithuania
Renata Macaitienė: Institute of Regional Development, Šiauliai Academy, Vilnius University, Vytauto Str. 84, LT-76352 Šiauliai, Lithuania

Mathematics, 2023, vol. 11, issue 4, 1-13

Abstract: 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 → ∞ .

Keywords: Epstein zeta-function; limit theorem; weak convergence; Haar measure (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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