Startseite On the value-distribution of iterated integrals of the logarithm of the Riemann zeta-function I: Denseness
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On the value-distribution of iterated integrals of the logarithm of the Riemann zeta-function I: Denseness

  • Kenta Endo und Shōta Inoue EMAIL logo
Veröffentlicht/Copyright: 7. Oktober 2020

Abstract

We consider iterated integrals of logζ(s) on certain vertical and horizontal lines. Here, the function ζ(s) is the Riemann zeta-function. It is a well-known open problem whether or not the values of the Riemann zeta-function on the critical line are dense in the complex plane. In this paper, we give a result for the denseness of the values of the iterated integrals on the horizontal lines. By using this result, we obtain the denseness of the values of 0tlogζ(12+it)𝑑t under the Riemann Hypothesis. Moreover, we show that, for any m2, the denseness of the values of an m-times iterated integral on the critical line is equivalent to the Riemann Hypothesis.

MSC 2010: 11M06; 11M26

Communicated by Valentin Blomer


Award Identifier / Grant number: 19J11223

Funding statement: The second author is supported by Grant-in-Aid for JSPS Research Fellow (Grant No. 19J11223).

Acknowledgements

The authors would like to deeply thank Professor Kohji Matsumoto for many useful comments and suggestions. They would also like to thank Mr Masahiro Mine for useful discussion.

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Received: 2020-03-25
Revised: 2020-06-04
Published Online: 2020-10-07
Published in Print: 2021-01-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

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