The prime number theorem for Beurling's generalized numbers. New cases

Abstract

This paper provides new cases of the prime number theorem for Beurling’s generalized prime numbers. Let N be the distribution of a generalized number system and let π be the distribution of its primes. It is shown that N(x) = ax + O(x/ log x) (C), γ > 3/2, where (C) stands for the Cesaro sense, is sufficient for the prime number theorem to hold, i.e., π(x) ∼ x/ log x. The Cesaro asymptotic estimate explicitly means that ∫ x 1 N(t)− at t ( 1− t x )m dt = O ( x log x ) , for somem ∈ N. Therefore, it includes Beurling’s classical condition. We also show that under these conditions the Mobius function, associated to the generalized number system, has mean value equal to 0. The methods of this article are based on complex Tauberian theorems for local pseudo-function boundary behavior and arguments from the theory of asymptotic behavior of Schwartz distributions.