Levinson's method in short intervals and simple zeros of the zeta function

Abstract

Fix theta between 1/2 and 1. We localize Levinson's method, with Conrey's general operator polynomial Q, to the intervals (T, T+T^theta], for mollifiers of length T^nu with nu less than theta-1/2, by running Young's short proof of the mollified second moment with a weight supported on the window. On such windows the method counts distinct sign changes of Hardy's function, with lower density kappa = 1 - log(c)/R, for every Q with Q(x)+Q(1-x) constant. For operator polynomials of degree 201 we certify kappa above 0.7170 nu for nu between 0.0199 and 0.0999. Used as the odd-support input of the Hilbert-parity inequality (Q-S)(N-O) at least 2(N-S)^2, together with Wang's short-interval pair-correlation theorem, this gives a positive proportion of simple critical zeros in (T, T+T^theta] for every fixed theta in [0.534, 1), compared with the previous exponents 0.552, 0.55019 and 0.5458838. At theta = 0.5459 the lower density exceeds 0.017763, about one thousand times the previous bound. Exact rational and ball arithmetic certify all constants, with an independent validated-quadrature replay. The results are asymptotic, use Wang's recent preprint for the pair term, and do not prove the Riemann hypothesis.