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On the Topology of the Sets of the Real Projections of the Zeros of Exponential Polynomials

  • Gaspar MoraEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 80)

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 \(1+\sum _{j=1}^{n}m_{j}\,\mathrm{{e}}^{\omega _{j}z}\), where \(n\) is a positive integer, \(m_{j}\in \mathbb {C}\setminus \left\{ 0\right\} \) and \(\omega _{j}>0\), for all \(1\le j\le n\). It is pointed out the influence of the \(\omega \)’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 \(\zeta _{n}(z):=\sum _{k=1}^{n}1/k^{z}\), \(n>2\), 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 \(n>2\).

Keywords

Exponential polynomials Zeros of the partial sums of the Riemann zeta function Perfect sets 

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of Mathematical AnalysisUniversity of AlicanteAlicanteSpain

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