On General Prime Number Theorems with Remainder

TL;DR

This paper establishes the equivalence between prime number theorems with remainders and integer counting functions in the context of Beurling generalized numbers, correcting past mistakes and providing an average version in Cesàro sense.

Contribution

It proves the equivalence of prime number theorems with remainders and integer counting functions for Beurling numbers, correcting earlier errors and introducing an average Cesàro version.

Findings

Equivalence of prime number theorem with remainders and integer counting functions.

Correction of mistakes in Nyman’s earlier work.

Introduction of an average Cesàro version of the theorem.

Abstract

We show that for Beurling generalized numbers the prime number theorem in remainder form $$\pi(x) = \operatorname*{Li}(x) + O\left(\frac{x}{\log^{n}x}\right) \quad \mbox{for all } n\in\mathbb{N}$$ is equivalent to (for some $a>0$) $$N(x) = ax + O\left(\frac{x}{\log^{n}x}\right) \quad \mbox{for all } n \in \mathbb{N},$$ where $N$ and $\pi$ are the counting functions of the generalized integers and primes, respectively. This was already considered by Nyman (Acta Math. 81 (1949), 299-307), but his article on the subject contains some mistakes. We also obtain an average version of this prime number theorem with remainders in the Ces\`aro sense.