JOINT DISCRETE APPROXIMATION OF ANALYTIC FUNCTIONS BY SHIFTS OF LERCH ZETA-FUNCTIONS

Abstract

The Lerch zeta-function $L(\lambda, \alpha,s)$, $s=\sigma+it$, depends on two real parameters $\lambda$ and $0<\alpha\leqslant 1$, and, for $\sigma>1$, is defined by the Dirichlet series $\sum_{m=0}^\infty \ee^{2\pi i\lambda m} (m+\alpha)^{-s}$, and by analytic continuation elsewhere. In the paper, we consider the joint approximation of collections of analytic functions by discrete shifts $(L(\lambda_1, \alpha_1, s+ikh_1), \dots, L(\lambda_r, \alpha_r, s+ikh_r))$, $k=0, 1, \dots$, with arbitrary $\lambda_j$, $0<\alpha_j\leqslant 1$ and $h_j>0$, $j=1, \dots, r$. We prove that there exists a non-empty closed set of analytic functions on the critical strip $1/2<\sigma<1$ which is approximated by the above shifts. It is proved that the set of shifts approximating a given collection of analytic functions has a positive lower density. The case of positive density also is discussed. A generalization for some compositions is given.