Number Theory Seminar
Wednesday, February 26, 2020, 3:15pm, 119 Physics
Fatma Cicek (University of Rochester)
A discrete analogue to Selberg's central limit theorem

Abstract:

Selberg's central limit theorem is one of the most significant probabilistic results in analytic number theory. Roughly, it states that the logarithm of the Riemann zeta-function $\zeta(s)$ on and near the critical line has an approximate two-dimensional Gaussian distribution. This talk will include a summary of the important ideas used in the proof of Selberg's central limit theorem. We will then present a discrete analogue for the sequences $\log\zeta(\rho+z)$ and $\log|\zeta'(\rho)|$, where $\rho$ denotes a nontrivial zero of the zeta-function and $z\neq 0$ is sufficiently small. Under suitable hypotheses like the Riemann Hypothesis and a zero-spacing hypothesis, our results show that these sequences also have an approximate Gaussian distribution. To give more perspective, we will mention generalizations of these results and finalize the talk with an application of Selberg's theorem to the zeros of linear combinations of Dirichlet $L$-functions.

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