| Title |
Application of stochastic elements in the universality of the periodic zeta-function: the case of short intervals |
| Authors |
Grigaliūnas, Marius ; Laurinčikas, Antanas ; Šiaučiūnas, Darius |
| DOI |
10.3390/axioms15010058 |
| Full Text |
|
| Is Part of |
Axioms.. Basel : MDPI. 2026, vol. 15, iss. 1, art. no. 58, p. [1-23].. eISSN 2075-1680 |
| Keywords [eng] |
approximation of analytic functions ; limit theorem ; periodic zeta-function ; universality ; weak convergence of probability measures |
| Abstract [eng] |
Let 𝔞 ={𝑎𝑚 :𝑚 ∈ℕ} be a multiplicative periodic sequence of complex numbers. In this paper, we consider the approximation of analytic functions defined in the strip {𝑠 =𝜎 +𝑖𝑡 :1/2 <𝜎 <1} by shifts 𝜁(𝑠 +𝑖𝜏;𝔞) of the zeta-function defined, for 𝜎 >1, by 𝜁(𝑠;𝔞) =∑∞ 𝑚=1𝑎𝑚 𝑚−𝑠 and by analytic continuation elsewhere. Using stochastic techniques, we obtain that the set of the above shifts approximating a given analytic function has a positive lower density (or density with at most countably many exceptions) in the interval [𝑇,𝑇 +𝑉] with 𝑇23/70 ⩽𝑉 ⩽𝑇1/2 as 𝑇 →∞. The proofs are based on a limit theorem with an explicitly given limit probability measure in the space of analytic functions. |
| Published |
Basel : MDPI |
| Type |
Journal article |
| Language |
English |
| Publication date |
2026 |
| CC license |
|
