Counting primes by sums of frequencies

TL;DR

This paper establishes a new sequence related to prime counting, demonstrating its asymptotic equivalence to the prime-counting function and providing estimates for their relationship.

Contribution

It introduces a novel sequence whose sum approximates the prime-counting function and derives asymptotic and limit relations between them.

Findings

_n o 0 as n o \u221e

_n ext{ is asymptotically equivalent to } rac{\u221e ext{(prime counting function)}}{n}

lim_{n o \u221e} ig(rac{1}{a_n} - rac{n}{\u221e(n)}ig) ext{ exists}

Abstract

We introduce the sequence $(a_n) \subset (0,1]$ and prove that the asymptotic behaviour of $\sum_{k=1}^n a_k$ is the same than $\pi(n)$, the prime-counting function. We also obtain that $\pi(n) \sim n a_n$ and we estimate $\frac{1}{a_n}-\frac{n}{\pi(n)}$ showing that $\lim_{n \rightarrow \infty} \frac{1}{a_n}-\frac{n}{\pi(n)}$ is convergent.