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Negative moments of the Riemann zeta-function

  • Hung M. Bui and Alexandra Florea EMAIL logo
Published/Copyright: January 6, 2024

Abstract

Assuming the Riemann Hypothesis, we study negative moments of the Riemann zeta-function and obtain asymptotic formulas in certain ranges of the shift in ζ ( s ) . For example, integrating | ζ ( 1 2 + α + i t ) | - 2 k with respect to t from T to 2 T , we obtain an asymptotic formula when the shift α is roughly bigger than 1 log T and k < 1 2 . We also obtain non-trivial upper bounds for much smaller shifts, as long as log 1 α log log T . This provides partial progress towards a conjecture of Gonek on negative moments of the Riemann zeta-function, and settles the conjecture in certain ranges. As an application, we also obtain an upper bound for the average of the generalized Möbius function.

Award Identifier / Grant number: DMS-2101769

Funding statement: The second author gratefully acknowledges support from the National Science Foundation, grant DMS-2101769, while working on this paper.

Acknowledgements

The authors would like to thank Steve Gonek, Jon Keating and Nathan Ng for helpful conversations and comments, Kannan Soundararajan for pointing out a mistake in a previous version of this paper, as well as the referee for several useful suggestions.

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Received: 2023-02-13
Revised: 2023-09-21
Published Online: 2024-01-06
Published in Print: 2024-01-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

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