On the Approximation by Mellin Transform of the Riemann Zeta-Function

Korolev, Maxim; Laurinčikas, Antanas

doi:10.3390/axioms12060520

Open AccessArticle

On the Approximation by Mellin Transform of the Riemann Zeta-Function

by

Maxim Korolev

^1,†

and

Antanas Laurinčikas

^2,*,†

¹

Department of Number Theory, Steklov Mathematical Institute of Russian Academy of Sciences, Gubkina Str. 8, 119991 Moscow, Russia

²

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

^*

Author to whom correspondence should be addressed.

^†

These authors contributed equally to this work.

Axioms 2023, 12(6), 520; https://doi.org/10.3390/axioms12060520

Submission received: 30 April 2023 / Revised: 18 May 2023 / Accepted: 24 May 2023 / Published: 25 May 2023

Download Versions Notes

Abstract

This paper is devoted to the approximation of a certain class of analytic functions by shifts

Z (s + i τ)

,

τ \in R

, of the modified Mellin transform

Z (s)

of the square of the Riemann zeta-function

ζ (1 / 2 + i t)

. More precisely, we prove the existence of a closed non-empty set F such that there are infinitely many shifts

Z (s + i τ)

, which approximate a given analytic function from F with a given accuracy. In the proof, the weak convergence of measures in the space of analytic functions is applied. Then, the set F coincides with the support of a limit measure.

Keywords: limit theorem; Mellin transform; Riemann zeta-function; weak convergence

Share and Cite

MDPI and ACS Style

Korolev, M.; Laurinčikas, A. On the Approximation by Mellin Transform of the Riemann Zeta-Function. Axioms 2023, 12, 520. https://doi.org/10.3390/axioms12060520

AMA Style

Korolev M, Laurinčikas A. On the Approximation by Mellin Transform of the Riemann Zeta-Function. Axioms. 2023; 12(6):520. https://doi.org/10.3390/axioms12060520

Chicago/Turabian Style

Korolev, Maxim, and Antanas Laurinčikas. 2023. "On the Approximation by Mellin Transform of the Riemann Zeta-Function" Axioms 12, no. 6: 520. https://doi.org/10.3390/axioms12060520

APA Style

Korolev, M., & Laurinčikas, A. (2023). On the Approximation by Mellin Transform of the Riemann Zeta-Function. Axioms, 12(6), 520. https://doi.org/10.3390/axioms12060520

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

Article Menu

On the Approximation by Mellin Transform of the Riemann Zeta-Function

Abstract

Share and Cite

Article Metrics

Article Access Statistics

Further Information

Guidelines

MDPI Initiatives

Follow MDPI