On the zeros of Riemann's zeta-function

Thesis (Ph. D.)--University of Rochester. Department of Mathematics, 2017.This thesis has three parts. In the first part, we combine the mollifier method with a zero detection method of Atkinson to prove in a new way that a positive proportion of the nontrivial zeros of the Riemann zeta-function ζ(s) are on the critical line. One of the main ingredients of the proof is an estimate for a mollified fourth moment of ζ(1/2 + it). We deduce this estimate from the twisted fourth moment formula that has been recently developed by Hughes and Young. The second part of this thesis is concerned with bounding the number N(σ, T) of zeros of ζ(s) that have real parts > σ and imaginary parts between 0 and T. We prove a claim of Conrey that improves previous bounds for N(σ, T) due to Selberg and Jutila. We do this by constructing a new mollifier that allows us to apply Conrey’s technique of using Kloosterman sum estimates to deduce an asymptotic formula for the mollified second moment of ζ(σ + it) when σ is > 1/2 and close to 1/2 . In the third part of the thesis, we give a conditional proof of the equivalence of certain asymptotic formulas for (a) averages over intervals for the 2-point form factor F([character did not render], T) for the zeros of ζ(s), (b) the mean square of the logarithmic derivative of ζ(s), (c) a variance for the number of primes in short intervals, and (d) the number of pairs of zeros of ζ(s) with small gaps. The main result is a generalization of the fusion of a theorem of Goldston and a theorem of Goldston, Gonek, and Montgomery. We apply our result to deduce several consequences of the Alternative Hypothesis