Abstract
The Lerch zeta function is defined by a Dirichlet series depending on two fixed parameters. In the paper, we consider the approximation of analytic functions by discrete shifts of the Lerch zeta function, and we prove that, for arbitrary parameters and a step of arithmetic progression, there is a closed non-empty subset of the space of analytic functions defined in the critical strip such that its functions can be approximated by discrete shifts of the Lerch zeta function. The set of those shifts is infinite, and it has a positive density. For the proof, the weak convergence of probability measures in the space of analytic functions is applied.
Subject
General Mathematics,Engineering (miscellaneous),Computer Science (miscellaneous)
Reference40 articles.
1. Note sur la fonction K(w,x,s)=∑n⩾0exp{2πinx}(n+w)-s;Lerch;Acta Math.,1887
2. Untersuchung einer aus vier Elementen gebildeten Reihe;Lipschitz;J. Reine Angew. Math.,1889
3. Laurinčikas, A., and Garunkštis, R. (2002). The Lerch Zeta Function, Kluwer Academic Publishers.
4. Theorem on the “universality” of the Riemann zeta function;Voronin;Math. USSR Izv.,1975
5. Voronin, S.M. (1977). Analytic Properties of Arithmetic Objects. [Ph.D. Thesis, V.A. Steklov Math. Inst.].