A New Joint Limit Theorem of Bohr–Jessen Type for Epstein Zeta-Functions

Laurinčikas, Antanas; Macaitienė, Renata

doi:10.3390/sym17060814

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A New Joint Limit Theorem of Bohr–Jessen Type for Epstein Zeta-Functions

by

Antanas Laurinčikas

^1,†

and

Renata Macaitienė

^2,*,†

¹

Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Naugarduko str. 24, LT-03225 Vilnius, Lithuania

²

Institute of Regional Development, Šiauliai Academy, Vilnius University, Vytauto str. 84, LT-76352 Šiauliai, Lithuania

^*

Author to whom correspondence should be addressed.

^†

These authors contributed equally to this work.

Symmetry 2025, 17(6), 814; https://doi.org/10.3390/sym17060814

Submission received: 25 April 2025 / Revised: 19 May 2025 / Accepted: 21 May 2025 / Published: 23 May 2025

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Abstract

For

j = 1, \dots, r

, let

Q_{j}

be a positive definite

n_{j} \times n_{j}

matrix, and

ζ (s_{j}; Q_{j})

denote the corresponding Epstein zeta-function. In this paper, assuming that

n_{j} ⩾ 4

is even and

{\underset{̲}{x}}^{T} Q_{j} \underset{̲}{x} \in Z

,

\underset{̲}{x} \in Z^{r} ∖ {\underset{̲}{0}}

, a joint limit theorem of Bohr–Jessen type for the functions

ζ (s_{1}; Q_{1}), \dots, ζ (s_{r}; Q_{r})

, by using generalizing shifts

ζ (σ_{1} + i φ_{1} (t); Q_{1}), \dots, ζ (σ_{r} + i φ_{r} (t); Q_{r})

, is proved. Here, the functions

φ_{1} (t), \dots, φ_{r} (t)

are increasing to

+ \infty

, with monotonic derivatives

φ_{j}^{'} (t)

satisfying the asymptotic growth conditions:

φ_{j} (t) ≪ t φ_{j}^{'} (t)

, and

φ_{j}^{'} (t) = o (φ_{j + 1}^{'} (t))

as

t \to \infty

. An explicit form of the limit measure is given. This theorem extends and generalizes the previous result on the joint value-distribution of Epstein zeta-functions.

Keywords: Epstein zeta-function; Haar measure; probability measures; weak convergence

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MDPI and ACS Style

Laurinčikas, A.; Macaitienė, R. A New Joint Limit Theorem of Bohr–Jessen Type for Epstein Zeta-Functions. Symmetry 2025, 17, 814. https://doi.org/10.3390/sym17060814

AMA Style

Laurinčikas A, Macaitienė R. A New Joint Limit Theorem of Bohr–Jessen Type for Epstein Zeta-Functions. Symmetry. 2025; 17(6):814. https://doi.org/10.3390/sym17060814

Chicago/Turabian Style

Laurinčikas, Antanas, and Renata Macaitienė. 2025. "A New Joint Limit Theorem of Bohr–Jessen Type for Epstein Zeta-Functions" Symmetry 17, no. 6: 814. https://doi.org/10.3390/sym17060814

APA Style

Laurinčikas, A., & Macaitienė, R. (2025). A New Joint Limit Theorem of Bohr–Jessen Type for Epstein Zeta-Functions. Symmetry, 17(6), 814. https://doi.org/10.3390/sym17060814

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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