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  1. ; ; (, Bulletin of the London Mathematical Society)
    Abstract We obtain conditional upper bounds for negative discrete moments of the derivative of the Riemann zeta‐function averaged over a subfamily of zeros of the zeta function that is expected to be arbitrarily close to full density inside the set of all zeros. For , our bounds for the ‐th moments are expected to be almost optimal. Assuming a conjecture about the maximum size of the argument of the zeta function on the critical line, we obtain upper bounds for these negative moments of the same strength while summing over a larger subfamily of zeta zeros. 
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  2. ; ; (, Mathematics of Computation)
    Assuming the Riemann hypothesis, we improve the current upper and lower bounds for the average value of Montgomery’s function F(a,T) over long intervals by means of a Fourier optimization framework. The function F(a,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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    Free, publicly-accessible full text available January 1, 2026