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

Author

Listed:
  • Antanas Laurinčikas

    (Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Naugarduko Str. 24, LT-03225 Vilnius, Lithuania
    These authors contributed equally to this work.)

  • Renata Macaitienė

    (Institute of Regional Development, Šiauliai Academy, Vilnius University, Vytauto Str. 84, LT-76352 Šiauliai, Lithuania
    These authors contributed equally to this work.)

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 → ∞ .

Suggested Citation

  • Antanas Laurinčikas & Renata Macaitienė, 2023. "A Generalized Discrete Bohr–Jessen-Type Theorem for the Epstein Zeta-Function," Mathematics, MDPI, vol. 11(4), pages 1-13, February.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:4:p:799-:d:1057939
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    References listed on IDEAS

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    1. Darius Šiaučiūnas & Raivydas Šimėnas & Monika Tekorė, 2021. "Approximation of Analytic Functions by Shifts of Certain Compositions," Mathematics, MDPI, vol. 9(20), pages 1-11, October.
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