Fourier optimization and Montgomery’s pair correlation conjecture

Carneiro, Emanuel; Milinovich, Micah; Ramos, Antonio Pedro

doi:10.1090/mcom/3990

Fourier optimization and Montgomery’s pair correlation conjecture
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by Emanuel Carneiro, Micah B. Milinovich and Antonio Pedro Ramos;

Math. Comp. 94 (2025), 409-424

DOI: https://doi.org/10.1090/mcom/3990

Published electronically: June 6, 2024

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Abstract:

Assuming the Riemann hypothesis, we improve the current upper and lower bounds for the average value of Montgomery’s function $F(\alpha , T)$ over long intervals by means of a Fourier optimization framework. The function $F(\alpha , T)$ is often used to study the pair correlation of the non-trivial zeros of the Riemann zeta-function. Two ideas play a central role in our approach: (i) the introduction of new averaging mechanisms in our conceptual framework and (ii) the full use of the class of test functions introduced by Cohn and Elkies for the sphere packing bounds, going beyond the usual class of bandlimited functions. We conclude that such an average value, that is conjectured to be $1$, lies between $0.9303$ and $1.3208$. Our Fourier optimization framework also yields an improvement on the current bounds for the analogous problem concerning the non-trivial zeros in the family of Dirichlet $L$-functions.

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Mathematics of Computation

Fourier optimization and Montgomery’s pair correlation conjecture
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Abstract: