On PNT equivalences for Beurling numbers

Abstract

In classical prime number theory several asymptotic relations are considered to be "equivalent" to the prime number theorem. In the setting of Beurling generalized numbers, this may no longer be the case. Under additional hypotheses on the generalized integer counting function, one can however still deduce various equivalences between the Beurling analogues of the classical PNT relations. We establish some of the equivalences under weaker conditions than were known so far.

21 pages

References in corpus (1)

• Some examples in the theory of Beurling's generalized prime numbers

Cited by in corpus (7)

• Complex Tauberian theorems for Laplace transforms with local pseudofunction boundary behavior
• Distribution of Beurling primes and zeroes of the Beurling zeta function I. Distribution of the zeroes of the zeta function of Beurling
• A Riemann-von Mangoldt-type formula for the distribution of Beurling primes
• On Diamond's $L^1$ criterion for asymptotic density of Beurling generalized integers
• Generalization of the Wiener-Ikehara theorem
• $M(x)=o(x)$ Estimates for Beurling numbers
• On the estimate $M(x)=o(x)$ for Beurling generalized numbers