Zeta statistics at the square-root barrier for cubes

Abstract

Hooley and Heath-Brown bounded, optimally up to X^ε , the number of integral zeros of ∑(i=1 to 6) x_i ^ 3 in [−X, X]^6, assuming the Riemann Hypothesis for geometric L-functions. We attribute the ε to several factors in harmonic analysis and algebraic geometry. We then remove the ε assuming, mainly, the Ratios Conjecture for geometric families of L-functions. We give applications to sums of three cubes.