On the q-analogue of the pair correlation conjecture via Fourier optimization
Abstract
We study the q-analogue of the average of Montgomery's function F(α, 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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