"Almost" universality of the Lerch zeta-function

The Lerch zeta-function L(lambda, alpha, s) with a transcendental parameter a, or with rational parameters alpha and lambda is universal, i.e., a wide class of analytic functions is approximated by shifts L(lambda, alpha, s + i tau), tau is an element of R. The case of an algebraic irrational a is an open problem. In the paper, it is proved that for all parameters alpha, 0 < alpha < 1, and lambda, 0 < lambda <= 1, including an algebraic irrational alpha, there exists a closed non-empty set of analytic functions F-alpha(,lambda) such that every function f is an element of F-alpha,F-lambda can be approximated by shifts L(lambda, alpha, s + i tau).