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On the distribution of zeros of derivatives of the Riemann ξ-function

On the distribution of zeros of derivatives of the Riemann ξ-function AbstractFor the completed Riemann zeta function ξ⁢(s){\xi(s)}, it is known that the Riemann hypothesis for ξ⁢(s){\xi(s)}implies the Riemann hypothesis for ξ(m)⁢(s){\xi^{(m)}(s)}, where m is any positive integer.In this paper, we investigate the distribution of the fractional parts of the sequence (α⁢γm){(\alpha\gamma_{m})}, where α is any fixed non-zero real number and γm{\gamma_{m}}runs over the imaginary parts of the zeros of ξ(m)⁢(s){\xi^{(m)}(s)}.We also obtain a zero density estimate and an explicit formula for the zeros of ξ(m)⁢(s){\xi^{(m)}(s)}.In particular, all our results hold uniformly for 0≤m≤g⁢(T){0\leq m\leq g(T)}, where the function g⁢(T){g(T)}tends to infinity with T and g⁢(T)=o⁢(log⁡log⁡T){g(T)=o(\log\log T)}. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

On the distribution of zeros of derivatives of the Riemann ξ-function

Forum Mathematicum , Volume 32 (1): 22 – Jan 1, 2020

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References (36)

Publisher
de Gruyter
Copyright
© 2019 Walter de Gruyter GmbH, Berlin/Boston
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/forum-2018-0081
Publisher site
See Article on Publisher Site

Abstract

AbstractFor the completed Riemann zeta function ξ⁢(s){\xi(s)}, it is known that the Riemann hypothesis for ξ⁢(s){\xi(s)}implies the Riemann hypothesis for ξ(m)⁢(s){\xi^{(m)}(s)}, where m is any positive integer.In this paper, we investigate the distribution of the fractional parts of the sequence (α⁢γm){(\alpha\gamma_{m})}, where α is any fixed non-zero real number and γm{\gamma_{m}}runs over the imaginary parts of the zeros of ξ(m)⁢(s){\xi^{(m)}(s)}.We also obtain a zero density estimate and an explicit formula for the zeros of ξ(m)⁢(s){\xi^{(m)}(s)}.In particular, all our results hold uniformly for 0≤m≤g⁢(T){0\leq m\leq g(T)}, where the function g⁢(T){g(T)}tends to infinity with T and g⁢(T)=o⁢(log⁡log⁡T){g(T)=o(\log\log T)}.

Journal

Forum Mathematicumde Gruyter

Published: Jan 1, 2020

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