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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. 

Understanding local hydraulic conditions is imperative to coastal harmful algal bloom (HAB) monitoring. The research summarized herein describes how the locations and tidal phases selected for coastal hazard sampling can influence measurement results used to guide management decisions for HABs. Our study was conducted in Frenchman Bay, Maine, known for its complex deglaciated coastline, strong tidal influence, and shellfishing activities that are susceptible to problematic HABs such as those produced by some species (spp.) of the diatom genus Pseudo-nitzschia. In-situ measurements of current velocity, density, and turbulence collected over a semidiurnal tidal cycle and a companion numerical model simulation of the study area provide concurrent evidence of two adjacent counter-rotating sub-mesoscale eddies (2–4 km diameter) that persist in the depth-averaged residual circulation. The eddies are generated in the wake of several islands in an area with abrupt bathymetric gradients, both legacy conditions partly derived from deglaciation ∼15 kya. Increased concentrations of Pseudo-nitzschia spp. measured during the semidiurnal survey follow a trend of elevated turbulent dissipation rates near the water surface, indicating that surface sampling alone might not adequately indicate species abundance. Additional measurements of Pseudo-nitzschia spp. from two years of weekly sampling in the region show that algal cell abundance is highest where residual eddies form. These findings provide incentive to examine current practices of HAB monitoring and management by linking coastal geomorphology to hydraulic conditions influencing HAB sampling outcomes, coastal morphometric features to material accumulation hotspots, and millennial time scales to modern hydraulic conditions.