  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.
