Richter, Florian Karl2021-11-262021-11-262021-11-262021-1010.1112/blms.12503https://infoscience.epfl.ch/handle/20.500.14299/1832392002.03255Let $\Omega(n)$ 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.text::journal::journal article::research article