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Density Intervals of Zeros of the Partial Sums of the Dirichlet Eta Function

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Abstract

It is shown that the set \(R_{n}:=\overline{\{ \mathfrak {R}z:\eta _{n}(z)=0\} }\) contains an interval \([ \alpha _{n},b_{n}] \) for some \(\alpha _{n}<0\) and \(0<b_{n}:=\sup \{ \mathfrak {R}z:\eta _{n}(z)=0\} \), where \(\eta _{n}(z):=\sum _{j=1}^{n}(-1)^{j-1}/j^{z}\) is the \(n^{\text {th}}\), \(n>2\), partial sum of the Dirichlet eta function \(\eta (z):=\sum _{j=1}^{\infty }(-1)^{j-1}/j^{z}\). It means that in the strip \([ \alpha _{n},b_{n}] \times \mathbb {R}\) no vertical sub-strip is zero-free for \(\eta _{n}(z)\), \(n>2\). Since \(\liminf _ {n\rightarrow \infty }b_{n}\ge 1\), that property is, in particular, asymptotically true for the partial sums \(\eta _{n}(z)\) in the critical strip \(( 0,1) \times \mathbb {R}\).

Keywords

Zeros of the partial sums of the eta function Exponential polynomials Diophantine approximation 

Mathematics Subject Classification

Primary: 11M26 Secondary: 11Lxx 11Jxx 

Notes

Acknowledgements

This work was partially supported by a grant from Ministerio de Economía y Competitividad, Spain (MTM 2014-52865-P).

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Facultad de Ciencias IIUniversidad de Alicante-Departamento de MatemáticasAlicanteSpain

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