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

• Gaspar Mora
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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© Springer International Publishing Switzerland 2014

## Authors and Affiliations

1. 1.Department of Mathematical AnalysisUniversity of AlicanteAlicanteSpain