On Self-Approximation of the Riemann Zeta Function in Short Intervals

Balčiūnas, Aidas; Laurinčikas, Antanas; Šiaučiūnas, Darius

doi:10.3390/sym17122075

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On Self-Approximation of the Riemann Zeta Function in Short Intervals

by

Aidas Balčiūnas

^†

,

Antanas Laurinčikas

^*,†

and

Darius Šiaučiūnas

^†

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

^*

Author to whom correspondence should be addressed.

^†

The authors contributed equally to this work.

Symmetry 2025, 17(12), 2075; https://doi.org/10.3390/sym17122075

Submission received: 28 October 2025 / Revised: 16 November 2025 / Accepted: 1 December 2025 / Published: 3 December 2025

(This article belongs to the Section Mathematics)

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Abstract

The Riemann hypothesis (RH) says that all zeros of the Riemann zeta function

ζ (s)

,

s = σ + i t

, in the strip

{s \in C : 0 < σ < 1}

lie on the line

σ = 1 / 2

. There are many equivalents of RH in various terms. In this paper, we propose equivalents of RH in terms of self-approximation, i.e., of the approximation of

ζ (s)

by

ζ (s + i τ)

,

τ \in R

, in the interval

τ \in [T, T + U]

with

T^{η} ⩽ U ⩽ T

,

η = 1273 / 4033

. We show that the RH is equivalent to the positivity of lower density and (with some exception for the accuracy of approximation) the density of the set of approximating shifts

ζ (s + i τ)

. For the proof, a probabilistic approach and mean square estimates for

ζ (s)

in short intervals are applied.

Keywords: equivalent of the Riemann hypothesis; limit theorem; non-trivial zeros; Riemann hypothesis; Riemann zeta function; universality; weak convergence of probability measures; zero-free region

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MDPI and ACS Style

Balčiūnas, A.; Laurinčikas, A.; Šiaučiūnas, D. On Self-Approximation of the Riemann Zeta Function in Short Intervals. Symmetry 2025, 17, 2075. https://doi.org/10.3390/sym17122075

AMA Style

Balčiūnas A, Laurinčikas A, Šiaučiūnas D. On Self-Approximation of the Riemann Zeta Function in Short Intervals. Symmetry. 2025; 17(12):2075. https://doi.org/10.3390/sym17122075

Chicago/Turabian Style

Balčiūnas, Aidas, Antanas Laurinčikas, and Darius Šiaučiūnas. 2025. "On Self-Approximation of the Riemann Zeta Function in Short Intervals" Symmetry 17, no. 12: 2075. https://doi.org/10.3390/sym17122075

APA Style

Balčiūnas, A., Laurinčikas, A., & Šiaučiūnas, D. (2025). On Self-Approximation of the Riemann Zeta Function in Short Intervals. Symmetry, 17(12), 2075. https://doi.org/10.3390/sym17122075

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On Self-Approximation of the Riemann Zeta Function in Short Intervals

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