On the 𝑞-analogue of the Pair Correlation Conjecture via Fourier optimization

Quesada-Herrera, Emily

doi:10.1090/mcom/3747

On the $q$-analogue of the Pair Correlation Conjecture via Fourier optimization
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by Emily Quesada-Herrera;

Math. Comp. 91 (2022), 2347-2365

DOI: https://doi.org/10.1090/mcom/3747

Published electronically: June 14, 2022

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Abstract:

We study the $q$-analogue of the average of Montgomery’s function $F(\alpha ,\, T)$ over bounded intervals. Assuming the Generalized Riemann Hypothesis for Dirichlet $L$-functions, we obtain upper and lower bounds for this average over an interval that are quite close to the pointwise conjectured value of $1$. To compute our bounds, we extend a Fourier analysis approach by Carneiro, Chandee, Chirre, and Milinovich, and apply computational methods of non-smooth programming.

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