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

  • Amita Malik and Arindam Roy EMAIL logo
Published/Copyright: September 6, 2019

Abstract

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

MSC 2010: 11M26; 11K38

Communicated by Freydoon Shahidi


Acknowledgements

The authors thank the referee for valuable comments and corrections which have improved the quality of this paper. The first author would also like to thank the Mathematics Department at Rice University for their hospitality where part of this research was completed.

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Received: 2018-04-12
Revised: 2018-10-07
Published Online: 2019-09-06
Published in Print: 2020-01-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

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