Almost Periodic Functions in terms of Bohr's Equivalence Relation

Abstract

In this paper we introduce an equivalence relation on the classes of almost periodic functions of a real or complex variable which is used to refine Bochner's result that characterizes these spaces of functions. In fact, with respect to the topology of uniform convergence, we prove that the limit points of the family of translates of an almost periodic function are precisely the functions which are equivalent to it, which leads us to a characterization of almost periodicity. In particular we show that any exponential sum which is equivalent to the Riemann zeta function, $ζ(s)$, can be uniformly approximated in $\{s=σ+it:σ>1\}$ by certain vertical translates of $ζ(s)$.

This is a modified version of our homonymous paper published in Ramanujan J. (2018). As we pointed out in our corrigendum (2019), we correct a gap found in the original version which is due to a property that does not generally hold, but it does under the assumption of existence of an integral basis. The earlier equivalence relation is revised to adapt correctly the situation in the general case. arXiv admin note: text overlap with arXiv:1711.04122