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Number Theory Seminar

Tuesday, November 8th, 2022

Time: 2:00 p.m.  Place: Jeffery Hall, Room 422

Speaker: Francesco Cellarosi (Queen’s University)

Title: The dynamical generalization of the Prime Number Theorem by Bergelson and Richter

Abstract: In a series of two talks, I will illustrate some of the results from the recent papers “Dynamical generalizations of the Prime Number Theorem and disjointness of additive and multiplicative semigroup actions” by V. Bergelson and F.K. Richter (Duke Math. J. 171(15): 3133-3200, October 2022) and “A new elementary proof of the Prime Number Theorem” by F.K. Richter (Bulletin of the London Math. Society. 53(5): 1365-1375, October 2021).

In the first talk, I will assume several results to give a proof of a generalization of the PNT. The key idea will be to replace the average of an arithmetic function by a double average. This will prove a uniform distribution for the sequence (T^\Omega(n)x)_{n\geq1}, where T is a uniquely ergodic transformation of a compact metric space X, x is a point in X, and \Omega(n) the number of prime factors of n (counted with multiplicity). A particular choice of X, T, and x will yield the classical PNT.

In the second talk, I will get into the proofs of the results used in the first talk. We will see that, in the proof of a key lemma, we could either use the PNT or a weaker form of the PNT. The latter choice yields a novel elementary proof of the PNT, along with several generalizations thereof (e.g. the PNT along arithmetic progressions).