Probabilistic number theory

During the fall semester, besides the Linear Algebra course for incoming Mathematics and Physics students, I was teaching a small course on Probabilistic Number Theory, or more precisely on a few aspects of probabilistic number theory that I find especially enjoyable.

Roughly speaking, I concentrated on a small number of examples of statements of convergence in law of sequences of arithmetically-defined probability measures: (1) the Erdös-Kac Theorem; (2) the Bohr-Jessen-Bagchi distribution theorems for the Riemann zeta function on the right of the critical line ; (3) Selberg’s Theorem on the normal behavior of the Riemann zeta function on the critical line ; (4) my own work with W. Sawin on Kloosterman paths. My goal in each case was to do with as little arithmetic, and as much probability, as possible. The reason for this is that I wanted to get and give as clear a picture as possible of which arithmetic facts are involved in each case.

This led to some strange and maybe pedantic-looking discussions at the beginning. But overall I think this was a useful point of view. I think it went especially well in the discussion of Bagchi’s Theorem, which concerns the “functional” limit theorem for vertical translates of the Riemann zeta function in the critical strip, but to the right of the critical line. The proof I gave is, I think, simpler and better motivated than those I have seen (all sources I know follow Bagchi very closely). In the section on Selberg’s Theorem, I used the recent proof found by Radziwill and Soundararajan, which is very elegant (although I didn’t have much time to cover full details during the class).

I have started writing notes for the course, which can be found here. Note that, as usual, the notes are potentially full of mistakes! But those parts which are written are fairly complete, both from the arithmetic and probabilistic point of view.

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Kowalski

I am a professor of mathematics at ETH Zürich since 2008.

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