A new elementary proof of the Prime Number Theorem

Abstract

Let $\Omega (n)$ denote the number of prime factors of $n$. We show that for any bounded $f:\mathbb{N}\to \mathbb{C}$ 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.

13 pages; to appear in Bulletin of the London Mathematical Society; corrected a small mistake in the proof of Proposition 2.2, see footnote 2 at the bottom of page 10

References in corpus (1)

• Dynamical generalizations of the Prime Number Theorem and disjointness of additive and multiplicative semigroup actions

Cited by in corpus (3)

• Dynamical generalizations of the Prime Number Theorem and disjointness of additive and multiplicative semigroup actions
• Dynamics on the number of prime divisors for additive arithmetic semigroups
• A New Elementary Proof of Landau's Prime Ideal Theorem, and Associated Results