A Generalized 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, 2022, vol. 10, issue 12, 1-11

Abstract: Let Q be a positive defined n × n matrix and Q [ x ̲ ] = x ̲ T Q x ̲ . 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 is meromorphically continued on the whole complex plane. Suppose that n ⩾ 4 is even and φ ( t ) is a differentiable function with a monotonic derivative. In the paper, it is proved that 1 T meas t ∈ [ 0 , T ] : ζ ( σ + i φ ( t ) ; Q ) ∈ A , A ∈ B ( C ) , converges weakly to an explicitly given probability measure on ( C , B ( C ) ) as T → ∞ .

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: 2022
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