Beurling Generalized Primes with Riemann Hypothesis and Beurling Generalized Primes with Large Oscillation

Abstract

The prime number theorem is central to analytic number theory. It has been conjectured for a century that the de la Vall?e Poussin remainder of the prime number theorem and the prime ideal theorem is essentially best possible. In a recent paper by H. Diamond, H. Montgomery and U. Vorhause an a following paper by the speaker, this conjecture has been settled down. In the paper of the speaker the main ideas of the first paper have been used in the study of the Riemann Hypothesis. In this talk, we will introduce the main ideas of the first paper [1] and some questions raised by the second paper [2]. The talk will show that the probabilistic method in the proof of the conjecture and the study of the Riemann Hypothesis are very interesting. References [1] Diamond, H., Montgomery, H., Vorhauer, U.: Beurling primes with large oscillation, Mathe. Annalen, 334, 1-36 (2006). [2] Zhang, W.-B.: Beurling primes with RH and Beurling primes with large oscillation, Math. Annalen, 337, 671-704 (2006).