A μ^{9/2} log μ upper bound for windowed Weil forms

Abstract

Let λmin(λ) be the bottom of the spectrum of Weil's quadratic form restricted to test functions supported in the multiplicative window [λ−1,λ], and put μ=λ2. By Weil's criterion, the Riemann Hypothesis is equivalent to λmin(λ) ≥ 0 for all λ. In previous papers we proved, without any hypothesis on the zeros of ζ, that λmin(λ) ≤ 1.437×1023 μ9/2(log μ)4e−4πμ for 50 ≤ μ ≤ 104. We locate the sources of the four logarithms: one is the density of the zeros of ζ at height ≈ μ, the other three are artefacts, removed by a translation in a Sobolev step and by keeping the exact Prüfer phase of the prolate functions and estimating a Poisson sum in L2 by non-stationary phase. We prove, unconditionally, λmin(λ) ≤ 3.50×1024 μ9/2 log μ e−4πμ for 50 ≤ μ ≤ 104, and λmin(λ) ≤ 7.58×1024 μ5(log μ)3e−4πμ for all μ ≥ 50. Numerical Galerkin computations of the even part at integer 7 ≤ μ ≤ 20 give ratios λmin/(1−λ4(2πμ)) between about 4.1 and 5.4 (not certified). The results are upper bounds only and say nothing about positivity.