A Generalization of Bohr’s Equivalence Theorem

Abstract

Based on a generalization of Bohr's equivalence relation for general Dirichlet series, in this paper we study the sets of values taken by certain classes of equivalent almost periodic functions in their strips of almost periodicity. In fact, the main result of this paper consists of a result like Bohr's equivalence theorem extended to the case of these functions.

A new geometrical perspective on Bohr-equivalence of exponential polynomials

“…It is worth noting the connection between Bohr's equivalence theorem [1, p.178] and part ii) of our next result (compare also with the results of the recent paper [10]).…”

“…Bohr used it in that case in order to get so-called Bohr's equivalence theorem, which shows that equivalent Dirichlet series take the same values in certain sets in the complex plane (e.g. see [1,11] and the recent paper [10]).…”

The Vortex-like Behavior of the Riemann Zeta Function to the Right of the Critical Strip

“…Based on the Bohr's equivalence relation, which was considered in [1, p. 173] for general Dirichlet series, we defined in [7][8][9][10] new equivalence relations in the more general context of the classes S Λ of exponential sums of type (1).…”

“…In this paper, we will use the following definition which constitutes the same equivalence relation as that of [9,Definition 2].…”

“…Throughout this work, we will use a generalization of Bohr's equivalence relation, defined in Sect. 2, which was used in [9] (see also [10]) to get a result like Bohr's equivalence theorem extended to certain classes of almost periodic functions in vertical strips {s ∈ C : α < Re s < β}.…”