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Zeroes of partial sums of the zeta-function

Part of: Zeta and $L$-functions: analytic theory Computational number theory

Published online by Cambridge University Press:  01 February 2016

David J. Platt  and
Timothy S. Trudgian
David J. Platt
Affiliation:
Heilbronn Institute for Mathematical Research , University of Bristol , Bristol , United Kingdom email dave.platt@bris.ac.uk
Timothy S. Trudgian
Affiliation:
Mathematical Sciences Institute , The Australian National University , ACT 0200 , Australia email timothy.trudgian@anu.edu.au

Abstract

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This article considers the positive integers $N$ for which ${\it\zeta}_{N}(s)=\sum _{n=1}^{N}n^{-s}$ has zeroes in the half-plane $\Re (s)>1$. Building on earlier results, we show that there are no zeroes for $1\leqslant N\leqslant 18$ and for $N=20,21,28$. For all other $N$ there are infinitely many such zeroes.

MSC classification

Primary: 11M06: $zeta (s)$ and $L(s, chi)$
Secondary: 11Y35: Analytic computations
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 19 , Issue 1 , 2016 , pp. 37 - 41
Copyright
© The Author(s) 2016 

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