Counting primes by sums of frequencies

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.