### References & Citations

# Mathematics > Number Theory

# Title: A new elementary proof of the Prime Number Theorem

(Submitted on 9 Feb 2020 (v1), last revised 3 Aug 2021 (this version, v3))

Abstract: Let $\Omega(n)$ denote the number of prime factors of $n$. We show that for any bounded $f\colon\mathbb{N}\to\mathbb{C}$ one has \[ \frac{1}{N}\sum_{n=1}^N\, f(\Omega(n)+1)=\frac{1}{N}\sum_{n=1}^N\, f(\Omega(n))+\mathrm{o}_{N\to\infty}(1). \] This yields a new elementary proof of the Prime Number Theorem.

## Submission history

From: Florian Karl Richter [view email]**[v1]**Sun, 9 Feb 2020 00:16:13 GMT (19kb)

**[v2]**Wed, 12 May 2021 19:26:23 GMT (18kb)

**[v3]**Tue, 3 Aug 2021 17:02:04 GMT (19kb)

Link back to: arXiv, form interface, contact.