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 paper shows that the conjecture of Lapidus and Van Frankenhuysen on the set of dimensions of fractality associated with a nonlattice fractal string is true in the important special case of a generic nonlattice self-similar string, but in general is false. The proof and the counterexample of this have been given by virtue of a result on exponential polynomials P (z), with real frequencies linearly independent over the rationals, that establishes a bound for the number of gaps of RP , the closure of the set of the real projections of its zeros, and the reason for which these gaps are produced.

Based on Bohr's equivalence relation which was established for general Dirichlet series, in this paper we introduce a new equivalence relation on the space of almost periodic functions in the sense of Besicovitch, B(R, C), defined in terms of polynomial approximations. From this, we show that in an important subspace B 2 (R, C) ⊂ B(R, C), where Parseval's equality and Riesz-Fischer theorem holds, its equivalence classes are sequentially compact and the family of translates of a function belonging to this subspace is dense in its own class.

This paper proves that the real projection of each simple zero of any partial sum of the Riemann zeta function n (s) := P n k=1 1 k s , n > 2, is an accumulation point of the set fRe s : n (s) = 0g.

Purpose -This paper aims to present a new method for obtaining points of the set determined by the closure of the real projections of the zeros of each partial sumof the Riemann zeta function and to show several applications of this result. Design/methodology/approach -The authors utilize an auxiliary function related to a known result of Avellar that characterizes the set of points of interest. Several figures and numerical experiences are presented to illustrate the various properties which are studied. Findings -It is first shown that each point of the image of the auxiliary function can be approximated by points of the image of the function formed by the approximants. Secondly, conditions are given on the auxiliary function to obtain points satisfying the property of density which is studied. Finally, by using these conditions, several useful applications are presented to the case n ¼ 4 and s 0 ¼ 0 which a more specific criterion is also given. Practical implications -This research is applicable for finding accumulation points of the set of the real projection of the zeros of the approximants on its critical interval. An exact interval included in this set is given for the case n ¼ 4. Also, it is demonstrated that the point 0 is included for a large set of values of n. Originality/value -The method employed is original and it contributes to the study on the properties of the density of the real parts of the zeros of a particular class of entire functions.