Papers by GASPAR MORA MARTINEZ
Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, 2012
This paper proves that the real projection of each simple zero of any partial sum of the Riemann ... more 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.
Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, 2014
In this paper we give an example of a nonlattice self-similar fractal string such that the set of... more In this paper we give an example of a nonlattice self-similar fractal string such that the set of real parts of their complex dimensions has an isolated point. This proves that, in general, the set of dimensions of fractality of a fractal string is not a perfect set.
Annali di Matematica Pura ed Applicata (1923 -), 2014
In this paper, we introduce a formula for the exact number of zeros of every partial sum of the R... more In this paper, we introduce a formula for the exact number of zeros of every partial sum of the Riemann zeta function inside infinitely many rectangles of the critical strips where they are situated.
Springer Proceedings in Mathematics & Statistics, 2014
ABSTRACT It is analysed some topological properties of the set P_{D(z)} of the real projections o... more ABSTRACT It is analysed some topological properties of the set P_{D(z)} of the real projections of the zeros of an exponential polynomial D(z) of the form 1+∑_{j=1}ⁿm_{j}e^{ω_{j}z}, where n is a positive integer, m_{j}∈ℂ∖{0} and ω_{j}>0, for all 1≤j≤n. It is pointed out the influence of the ω's, called frequencies, as well as the m's, called coefficients of D(z), on the topology of P_{D(z)}. Reciprocally, it is seen how the order of the zeros of D(z) is also influenced by the topology of P_{D(z)}. Finally, has been proved that the set of the real projections of the zeros of every partial sum ζ_{n}(z):=∑_{k=1}ⁿ1/k^{z}, n>2, of the Riemann zeta function situated in the critical strip of the Riemann zeta function is dense in itself. Therefore the closure of this set is a perfect set, for all n>2.
We approximate the multiple integral of a non-negative real function f of class C 1 on the unit c... more We approximate the multiple integral of a non-negative real function f of class C 1 on the unit cube [0, 1] n , n ≥ 2, by a simple one on the interval [−1, 1] by using a technique of densification of the region of quadrature. The curve that densifies is an α-dense curve called cosines curve. The integrand of the simple integral depends, as that of [8], on the Chebyshev polynomial of second kind. An estimation on the error generated by this reduction is also settled.
Mediterranean Journal of Mathematics, 2012
This paper proves that every zero of any n th , n 2, partial sum of the Riemann zeta function pro... more This paper proves that every zero of any n th , n 2, partial sum of the Riemann zeta function provides a vector space of basic solutions of the functional equation f (x) + f (2x) + ::: + f (nx) = 0, x 2 R. The continuity of the solutions depends on the sign of the real part of each zero.
Kybernetes, 2012
Purpose-In this article the aim is to propose a new form to densify parallelepipeds of R N by seq... more Purpose-In this article the aim is to propose a new form to densify parallelepipeds of R N by sequences of a-dense curves with accumulated densities. Design/methodology/approach-This will be done by using a basic a-densification technique and adding the new concept of sequence of a-dense curves with accumulated density to improve the resolution of some global optimization problems. Findings-It is found that the new technique based on sequences of a-dense curves with accumulated densities allows to simplify considerably the process consisting on the exploration of the set of optimizer points of an objective function with feasible set a parallelepiped K of R N. Indeed, since the sequence of the images of the curves of a sequence of a-dense curves with accumulated density is expansive, in each new step of the algorithm it is only necessary to explore a residual subset. On the other hand, since the sequence of their densities is decreasing and tends to zero, the convergence of the algorithm is assured. Practical implications-The results of this new technique of densification by sequences of a-dense curves with accumulated densities will be applied to densify the feasible set of an objective function which minimizes the quadratic error produced by the adjustment of a model based on a beta probability density function which is largely used in studies on the transition-time of forest vegetation. Originality/value-A sequence of a-dense curves with accumulated density represents an original concept to be added to the set of techniques to optimize a multivariable function by the reduction to only one variable as a new application of a-dense curves theory to the global optimization.
Kybernetes, 2012
PurposeThis paper aims to introduce a new class of entire functions whose zeros (zk)k≥1satisfy ∑k... more PurposeThis paper aims to introduce a new class of entire functions whose zeros (zk)k≥1satisfy ∑k=1∞Im zk=O(1).Design/methodology/approachThis is done by means of a Ritt's formula which is used to prove that every partial sum of the Riemann Zeta function,ζn(z):=∑k=1n1/kz,n≥2, has zeros (snk)k≥1verifying ∑k=1∞Re snk=O(1) and extending this property to a large class of entire functions denoted byAO.FindingsIt is found that this new classAOhas a part in common with the classAintroduced by Levin but is distinct from it. It is shown that, in particular,AOcontains every partial sum of the Riemann Zeta functionζn(iz) and every finite truncation of the alternating Dirichlet series expansion of the Riemann zeta function,Tn(iz):=∑k=1n(−1)k−1/kiz, for alln≥2.Practical implicationsWith the exception of then=2 case, numerical experiences show that all zeros ofζn(z) andTn(z) are not symmetrically distributed around the imaginary axis. However, the fact consisting of every functionζn(iz) andTn...
Journal of Mathematical Analysis and Applications, 2009
In this paper we study the distribution of zeros of each entire function of the sequence {G n (z)... more In this paper we study the distribution of zeros of each entire function of the sequence {G n (z) ≡ 1 + 2 z + • • • + n z : n 2}, which approaches the Riemann zeta function for Re z < −1, and is closely related to the solutions of the functional equations f (z) + f (2z) + • • • + f (nz) = 0. We determine the density of the zeros of G n (z) on the critical strip where they are situated by using almost-periodic functions techniques. Furthermore, by using a theorem of Kronecker, we also establish a formula for the number of zeros of G n (z) inside certain rectangles in the critical strip.
Bulletin of the London Mathematical Society, 2013
This paper shows that the conjecture of Lapidus and Van Frankenhuysen on the set of dimensions of... more 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.
Abstract and Applied Analysis, 2011
We give a partition of the critical strip, associated with each partial sum of the Riemann zeta f... more We give a partition of the critical strip, associated with each partial sum of the Riemann zeta function for Re , 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.
Complex Analysis and Operator Theory, 2012
This paper shows, by means of Kronecker's theorem, the existence of infinitely many privileged re... more This paper shows, by means of Kronecker's theorem, the existence of infinitely many privileged regions called r-rectangles (rectangles with two semicircles of small radius r) in the critical strip of each function L n (z) := 1 − n k=2 k z , n ≥ 2, containing exactly T log n 2π + 1 zeros of L n (z), where T is the height of the r-rectangle and [•] represents the integer part.
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Papers by GASPAR MORA MARTINEZ