# On General Prime Number Theorems with Remainder

@inproceedings{Debruyne2017OnGP,
title={On General Prime Number Theorems with Remainder},
author={Gregory Debruyne and Jasson Vindas},
year={2017}
}
• Published 2017
• Mathematics
• We show that for Beurling generalized numbers the prime number theorem in remainder form $$\pi \left( x \right) = Li\left( x \right) + O\left( {\frac{x} {{\log ^n x}}} \right)\,for\,all\,n\, \in \,{\Bbb N}$$ is equivalent to (for some a > 0) $$N\left( x \right) = ax + O\left( {\frac{x} {{\log ^n x}}} \right)\,for\,all\,n\, \in \,{\Bbb N}$$ where N and π are the counting functions of the generalized integers and primes, respectively. This was already considered by Nyman (Acta… CONTINUE READING

Create an AI-powered research feed to stay up to date with new papers like this posted to ArXiv

## Complex Tauberian theorems and applications to Beurling generalized primes

VIEW 1 EXCERPT
CITES BACKGROUND

#### References

##### Publications referenced by this paper.
SHOWING 1-10 OF 18 REFERENCES

## Analyse de la loi asymptotique de la distribution des nombres premiers généralisés. I

VIEW 6 EXCERPTS
HIGHLY INFLUENTIAL

## A distributional approach to asymptotics

• Mathematics
• 2002
VIEW 4 EXCERPTS
HIGHLY INFLUENTIAL

## Some examples in the theory of Beurling’s generalized prime numbers

• Mathematics
• 2016
VIEW 1 EXCERPT

## Extensions of Beurling's prime number theorem

VIEW 1 EXCERPT

## Asymptotic distribution of integers with certain prime factorizations

• Mathematics
• 2014
VIEW 1 EXCERPT

## Asymptotic Behavior of Generalized Functions

• Mathematics, Computer Science
• Series on Analysis, Applications and Computation
• 2011
VIEW 3 EXCERPTS

## The structure of quasiasymptotics of Schwartz distributions

VIEW 1 EXCERPT

## Structural theorems for quasiasymptotics of distributions at infinity

VIEW 1 EXCERPT

## Sur les nombres premiers généralisés de Beurling. Preuve d'une conjecture de Bateman et Diamond

VIEW 1 EXCERPT