The second moment of the Riemann zeta function at its local extrema

Andrew E B S Pearce-Crump; Christopher Hughes; Solomon Lugmayer

doi:10.1112/jlms.70250

The second moment of the Riemann zeta function at its local extrema

Andrew E B S Pearce-Crump
, Christopher Hughes^*
, Solomon Lugmayer

^*Corresponding author for this work

School of Mathematics

Research output: Contribution to journal › Article (Academic Journal) › peer-review

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Abstract

Conrey and Ghosh studied the second moment of the Riemann zeta function, evaluated at its local extrema along the critical line, finding the leading order behaviour to be $frac{e^2-5}{4 \pi} T (\log T)^2$. This problem is closely related to a mixed moment of the Riemann zeta function and its derivative. We present a new approach which will uncover the lower order terms for the second moment as a descending chain of powers of logarithms in the asymptotic expansion.

Original language	English
Article number	e70250
Journal	Journal of the London Mathematical Society
Volume	112
Issue number	2
Early online date	29 Jul 2025
DOIs	https://doi.org/10.1112/jlms.70250
Publication status	Published - 1 Aug 2025

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© 2025 The Author(s). The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.

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title = "The second moment of the Riemann zeta function at its local extrema",

abstract = "Conrey and Ghosh studied the second moment of the Riemann zeta function, evaluated at its local extrema along the critical line, finding the leading order behaviour to be \$frac\{e\textasciicircum{}2-5\}\{4 \textbackslash{}pi\} T (\textbackslash{}log T)\textasciicircum{}2\$. This problem is closely related to a mixed moment of the Riemann zeta function and its derivative. We present a new approach which will uncover the lower order terms for the second moment as a descending chain of powers of logarithms in the asymptotic expansion.",

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N2 - Conrey and Ghosh studied the second moment of the Riemann zeta function, evaluated at its local extrema along the critical line, finding the leading order behaviour to be $frac{e^2-5}{4 \pi} T (\log T)^2$. This problem is closely related to a mixed moment of the Riemann zeta function and its derivative. We present a new approach which will uncover the lower order terms for the second moment as a descending chain of powers of logarithms in the asymptotic expansion.

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The second moment of the Riemann zeta function at its local extrema

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