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

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Title: Density Intervals of Zeros of the Partial Sums of the Dirichlet Eta Function
Authors: Mora, Gaspar
Research Group/s: Curvas Alpha-Densas. Análisis y Geometría Local
Center, Department or Service: Universidad de Alicante. Departamento de Matemáticas
Keywords: Zeros of the partial sums of the eta function | Exponential polynomials | Diophantine approximation
Knowledge Area: Análisis Matemático
Issue Date: Dec-2018
Publisher: Springer International Publishing
Citation: Mediterranean Journal of Mathematics. 2018, 15:208. doi:10.1007/s00009-018-1252-3
Abstract: It is shown that the set Rn := {Rz : ηn(z) = 0} contains an interval [αn, bn] for some αn < 0 and 0 < bn := sup{Rz : ηn(z) = 0}, where ηn(z) := Σn j=1(−1)j−1/jz is the nth, n > 2, partial sum of the Dirichlet eta function η(z) := Σ∞ j=1(−1)j−1/jz. It means that in the strip [αn, bn]×R no vertical sub-strip is zero-free for ηn(z), n > 2. Since lim infn→∞ bn ≥ 1, that property is, in particular, asymptotically true for the partial sums ηn(z) in the critical strip (0, 1) × R.
Sponsor: This work was partially supported by a grant from Ministerio de Economía y Competitividad, Spain (MTM 2014-52865-P).
ISSN: 1660-5446 (Print) | 1660-5454 (Online)
DOI: 10.1007/s00009-018-1252-3
Language: eng
Type: info:eu-repo/semantics/article
Rights: © Springer Nature Switzerland AG 2018
Peer Review: si
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