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The fourth moment of the Hurwitz zeta function

  • Winston Heap ORCID logo EMAIL logo and Anurag Sahay ORCID logo
Published/Copyright: November 23, 2024

Abstract

We prove a sharp upper bound for the fourth moment of the Hurwitz zeta function ζ ( s , α ) on the critical line when the shift parameter 𝛼 is irrational and of irrationality exponent strictly less than 3. As a consequence, we determine the order of magnitude of the 2 k th moment for all 0 k 2 in this case. In contrast to the Riemann zeta function and other 𝐿-functions from arithmetic, these grow like T ( log T ) k . This suggests, and we conjecture, that the value distribution of ζ ( s , α ) on the critical line is Gaussian.

Funding statement: A. Sahay is partially supported by Purdue University start-up funding available to Trevor Wooley and by the AMS-Simons Travel Grant.

Acknowledgements

We thank Steve Gonek and Trevor Wooley for useful discussions on these topics. We also thank the anonymous referee for their comments and suggestions.

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Received: 2024-05-31
Published Online: 2024-11-23
Published in Print: 2025-01-01

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