Extension of a Bohr-Jessen type theorem for the Epstein zeta-function in short intervals.

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Bibliographic Details
Title:	Extension of a Bohr-Jessen type theorem for the Epstein zeta-function in short intervals.
Authors:	Balčiūnas, Aidas¹, Garbaliauskienė, Virginija², Macaitienė, Renata² renata.macaitiene@sa.vu.lt
Source:	Mathematical Modelling & Analysis. 2026, Vol. 31 Issue 2, p289-302. 15p.
Subjects:	Zeta functions, Limit theorems, Probability measures, Modular forms, Dirichlet series
Abstract:	The article focuses on extending a Bohr-Jessen type limit theorem for the Epstein zeta-function, a complex function defined via a positive definite matrix Q, to short intervals on the critical strip. Building on prior work that established weak convergence of probability measures associated with the value distribution of the Epstein zeta-function over long intervals, the authors prove that this convergence also holds when the interval length H satisfies \(H \geq T^{27/82}\) as \(T \to \infty\). The proof employs a representation of the Epstein zeta-function in terms of Dirichlet L-functions and modular forms, mean square estimates for Hurwitz zeta-functions in short intervals, and techniques from probability theory concerning weak convergence and Haar measures on compact groups. This result generalizes earlier limit theorems by demonstrating that the statistical distribution of values of the Epstein zeta-function remains stable even when restricted to significantly shorter intervals. [Extracted from the article]
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Description
Abstract:	The article focuses on extending a Bohr-Jessen type limit theorem for the Epstein zeta-function, a complex function defined via a positive definite matrix Q, to short intervals on the critical strip. Building on prior work that established weak convergence of probability measures associated with the value distribution of the Epstein zeta-function over long intervals, the authors prove that this convergence also holds when the interval length H satisfies \(H \geq T^{27/82}\) as \(T \to \infty\). The proof employs a representation of the Epstein zeta-function in terms of Dirichlet L-functions and modular forms, mean square estimates for Hurwitz zeta-functions in short intervals, and techniques from probability theory concerning weak convergence and Haar measures on compact groups. This result generalizes earlier limit theorems by demonstrating that the statistical distribution of values of the Epstein zeta-function remains stable even when restricted to significantly shorter intervals. [Extracted from the article]
ISSN:	13926292
DOI:	10.3846/mma.2026.24285