Abstract
In the paper, an universality theorem on the approximation of analytic functions by generalized discrete shifts of zeta functions of Hecke-eigen cusp forms is obtained. These shifts are defined by using the function having continuous derivative satisfying certain natural growth conditions and, on positive integers, uniformly distributed modulo 1.
Highlights
It turned out that some other zeta and L-functions are universal in the Voronin sense, among them, zeta-functions of certain cusp forms
In [16], he obtained a discrete universality theorem for Dedekind zeta-functions
The first discrete universality theorem for ζ(s, F ) attached to a new form F (z), under a certain arithmetical hypothesis for the number h, was proved in [9]
Summary
In [16], he obtained a discrete universality theorem for Dedekind zeta-functions. In his theorem, τ takes values from the arithmetic progression {kh: k ∈ N0 = N ∪ {0}}, where h > 0 is a fixed number. The first discrete universality theorem for ζ(s, F ) attached to a new form F (z), under a certain arithmetical hypothesis for the number h, was proved in [9]. The aim of this paper is to prove a discrete universality theorem for the function ζ(s, F ) when τ in ζ(s + iτ, F ) runs over some general sequence of real numbers. N : sup ζ s + iφ(k), F − f (s) < ε > 0 s∈K exists for all but at most countably many ε > 0
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