A new elementary proof of the Prime Number Theorem

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

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