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In the paper, we approximate analytic functions by generalized shifts of the Lerch zeta-function, where g is a certain increasing to real function having a monotonic derivative. We prove that, for arbitrary parameters λ and α, there exists a closed set  of analytic functions defined in the strip 1/2 < σ < 1 which functions are approximated by the above shifts. If the set of logarithms is linearly independent over the field of rational numbers, then the set  coincides with the set of all analytic functions in that strip.

The Lerch zeta-function depends on two real parameters λ and  and, for σ > 1, is defined by the Dirichlet series , and by analytic continuation elsewhere. In the paper, we consider the joint approximation of collections of analytic functions by discrete shifts  with arbitrary 1 and We prove that there exists a non-empty closed set of analytic functions on the critical strip 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.

In this paper, we consider the simultaneous approximation of tuples of analytic functions by tuples of shifts of Lerch zeta-functions with arbitrary parameters. We prove that there exists a closed set of tuples of functions analytic in the right-hand side of the critical strip, which is approximated by the above tuples of shifts. Further, a generalization for some compositions of tuples of Lerch zeta-functions is given.