Francesco Cellarosi (Queen鈥檚 University)
Date
Tuesday November 15, 20222:00 pm - 3:00 pm
Location
Jeffery Hall, Room 422Number Theory Seminar
Tuesday, November 15th, 2022
Time: 2:00 p.m. Place: Jeffery Hall, Room 422
Speaker: Francesco Cellarosi (Queen鈥檚 University)
Title: The dynamical generalization of the Prime Number Theorem by Bergelson and Richter (Part 2)
Abstract: In a series of two talks, I will illustrate some of the results from the recent papers 鈥淒ynamical generalizations of the Prime Number Theorem and disjointness of additive and multiplicative semigroup actions鈥 by V. Bergelson and F.K. Richter (Duke Math. J. 171(15): 3133-3200, October 2022) and 鈥淎 new elementary proof of the Prime Number Theorem鈥 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).