Lethbridge Number Theory and Combinatorics Seminar: Nathan Ng

The prime number theorem proven independently by de la Vallée Poussin and Hadamard (1896) is an asymptotic statement about prime counting functions. It holds for sufficiently large numbers x. In 1941 Rosser authored an article giving explicit versions of the prime number theorem which holds for x in various ranges. This work was later updated in 1962 and 1975, in joint work of Rosser and Schoenfeld. In recent years there has been a flurry of activity on this subject with contributions made by Dusart, Faber-Kadiri, Büthe, Fiori-Kadiri-Swidinsky, Johnston-Yang, and Chirre-Helfgott. Very recently, Tao has initiated the Integrated Explicit Analytic Number Theory network which has the goal to formalize many results in this field. Some of the key ideas that are used include a partial verification of the Riemann Hypothesis, explicit zero-free regions, explicit zero-density, explicit zero-counting formulae, and optimal functions. In this talk, I will provide a survey and history of results in explicit prime number theory. I will also present recent new bounds on Mertens sums and products which is joint work with Broadbent, Fiori, Kadiri, and Wilk.