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A discrete analogue to Selberg's central limit theorem

Duke Math
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Wednesday, February 26, 2020
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3:15 pm - 4:15 pm
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Fatma Cicek (University of Rochester)
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Number Theory Seminar

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 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 two logarithmic sequences. 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.