Juan Matias Sepulcre

The Critical Strips of the Sums

We give a partition of the critical strip, associated with each partial sum 1 + 2 𝑧 + ⋯ + 𝑛 𝑧 of the Riemann zeta function for Re 𝑧 − 1 , formed by infinitely many rectangles for which a formula allows us to count the number of its zeros inside each of them with an error, at most, of two zeros. A generalization of this formula is also given to a large class of almost-periodic functions with bounded spectrum.

On the Complex Dimensions of Nonlattice Fractal Strings in Connection with Dirichlet Polynomials

In this paper we give a new characterization of the closure of the set of the real parts of the zeros of a particular class of Dirichlet polynomials that is associated with the set of dimensions of fractality of certain fractal strings. We show, for some representative cases of nonlattice Dirichlet polynomials, that the real parts of their zeros are dense in their associated critical intervals, confirming the conjecture and the numerical experiments made by M. Lapidus and M. van Frankenhuysen in several papers.

On the Result of Invariance of the Closure Set of the Real Projections of the Zeros of an Important Class of Exponential Polynomials

We provide the proof of a practical pointwise characterization of the set defined by the closure set of the real projections of the zeros of an exponential polynomial with real frequencies linearly independent over the rationals. As a consequence, we give a complete description of the set and prove its invariance with respect to the moduli of the , which allows us to determine exactly the gaps of and the extremes of the critical interval of by solving inequations with positive real numbers. Finally, we analyse the converse of this result of invariance.

Equivalence classes of exponential polynomials with the same set of zeros

Through several equivalence binary relations, in this paper we identify, on the one hand, groups of exponential polynomials with the same set of zeros, and on the other hand, groups of functional equations of the form that lead to equivalent exponential polynomials with the same set of zeros.

A new approach to obtain points of the closure of the real parts of the zeros of the partial sums 1 + 2z +ċ+nz,n≥2

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 sum 1+2s+ċ+ns, n≥2, s=σ+it, of 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=4 and σ0=0 which a more specific criterion is also given.Practical implications – This researc...

Bohr’s equivalence relation in the space of Besicovitch almost periodic functions

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,

A class of functions whose sum of zeros has bounded real part

B(\mathbb {R},\mathbb {C})

On the distribution of zeros of a sequence of entire functions approaching the Riemann zeta function

B(R,C), defined in terms of polynomial approximations. From this, we show that in an important subspace