| Title |
On self-approximation of the Riemann zeta function in short intervals |
| Authors |
Balčiūnas, Aidas ; Laurinčikas, Antanas ; Šiaučiūnas, Darius |
| DOI |
10.3390/sym17122075 |
| Full Text |
|
| Is Part of |
Symmetry.. Basel : MDPI. 2025, vol. 17, iss. 12, art. no. 2075, p. [1-14].. eISSN 2073-8994 |
| Keywords [eng] |
equivalent of the Riemann hypothesis ; limit theorem ; non-trivial zeros ; Riemann hypothesis ; Riemann zeta function ; universality ; weak convergence of probability measures ; zero-free region |
| Abstract [eng] |
The Riemann hypothesis (RH) says that all zeros of the Riemann zeta function 𝜁(𝑠), 𝑠=𝜎+𝑖𝑡, in the strip {𝑠∈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+𝑖𝜏), 𝜏∈ℝ, in the interval 𝜏∈[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+𝑖𝜏). For the proof, a probabilistic approach and mean square estimates for 𝜁(𝑠) in short intervals are applied. |
| Published |
Basel : MDPI |
| Type |
Journal article |
| Language |
English |
| Publication date |
2025 |
| CC license |
|
