Chebyshevskii Sbornik
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
General information
Latest issue
Archive

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Chebyshevskii Sb.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Chebyshevskii Sbornik, 2018, Volume 19, Issue 1, Pages 138–151
DOI: https://doi.org/10.22405/2226-8383-2018-19-1-138-151
(Mi cheb627)
 

Joint discrete universality for Lerch zeta-functions

A. Laurinčikas, A. Mincevič

Vilnius University
References:
Abstract: After Voronin's work of 1975, it is known that some of zeta and $L$-functions are universal in the sense that their shifts approximate a wide class of analytic functions. Two cases of shifts, continuous and discrete, are considered.
The present paper is devoted to the universality of Lerch zeta-functions $L(\lambda, \alpha, s)$, $s= \sigma+it $, which are defined, for $ \sigma > 1$, by the Dirichlet series with terms $ e^{2 \pi i \lambda m}(m+ \alpha)^{-s} $ with parameters $\lambda \in \mathbb{R} $ and $\alpha$, $0 < \alpha \leqslant 1$, and by analytic continuation elsewhere. We obtain joint discrete universality theorems for Lerch zeta-functions. More precisely, a collection of analytic functions $ f_{1}(s), \dots, f_{r}(s) $ simultaneously is approximated by shifts $L(\lambda_{1},\alpha_{1},s+ikh),\dots, L(\lambda_{r},\alpha_{r},s+ikh)$, $k=0,1,2,\dots$, where $h>0$ is a fixed number. For this, the linear independence over the field of rational numbers for the set $\left \{ (\log (m+ \alpha_{j}): m \in \mathbb{N}_{0},\; j=1,\dots,r),\frac{2 \pi}{h} \right\}$ is required. For the proof, probabilistic limit theorems on the weak convergence of probability measures in the space of analytic function are applied.
Keywords: Lerch zeta-function, Mergelyan theorem, space of analytic functions, universality, weak convergence.
Funding agency Grant number
ESF - European Social Fund 09.3.3-LMT-K-712-01-0037
The research of the first author is funded by the European Social Fund according to the activity “Improvement of researchers” qualification by implementing world-class R&D projects' of Measure No. 09.3.3-LMT-K-712-01-0037.
Document Type: Article
UDC: 511.3
Language: Russian
Citation: A. Laurinčikas, A. Mincevič, “Joint discrete universality for Lerch zeta-functions”, Chebyshevskii Sb., 19:1 (2018), 138–151
Citation in format AMSBIB
\Bibitem{LauMin18}
\by A.~Laurin{\v{c}}ikas, A.~Mincevi{\v{c}}
\paper Joint discrete universality for Lerch zeta-functions
\jour Chebyshevskii Sb.
\yr 2018
\vol 19
\issue 1
\pages 138--151
\mathnet{http://mi.mathnet.ru/cheb627}
\crossref{https://doi.org/10.22405/2226-8383-2018-19-1-138-151}
Linking options:
  • https://www.mathnet.ru/eng/cheb627
  • https://www.mathnet.ru/eng/cheb/v19/i1/p138
  • Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Statistics & downloads:
    Abstract page:231
    Full-text PDF :59
    References:30
     
      Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2024