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 ${\lim }_{n\to \infty }\frac{1}{a_n}-\frac{n}{\pi (n)}$ is convergent.