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Thursday, September 7, 2017

**Abstract:** In the first part of this talk, we present a new proof that a positive proportion of the zeros of the Riemann zeta-function lie on the critical line. The proof is an enhancement of a zero-detection method of Atkinson from the 1940’s, and uses the recent estimate of Hughes and Young for the twisted fourth moment of zeta. In the second part, we consider the number of zeros of zeta inside the region with real part larger than $\sigma$ and imaginary part between 0 and T. A bound for this number is called a “zero-density estimate.” We present an improved zero-density estimate for the case when $\sigma$ is larger than 1/2 but close to 1/2. The main theorem confirms an unproved result of Conrey from the 1980’s using his technique of applying Kloosterman sum estimates. Finally, in the third part, we look at hypothetical statements for the vertical distribution of zeros along the critical line and deduce their consequences for the prime numbers and other properties of zeta. The main theorem generalizes results of Goldston, Gonek, and Montgomery that give consequences of the pair correlation conjecture. We apply the theorem to examine implications of the well-known “alternative hypothesis,” which is related to Landau-Siegel zeros.