An Exact Formula for the Prime Counting Function

TL;DR

This paper derives a precise formula for the prime counting function using advanced number theory concepts, avoiding reliance on the zeros of the zeta function, and introduces a powerful inversion formula for Dirichlet series.

Contribution

It presents a new exact formula for π(x) based on a generalized approach involving the Möbius function and Dirichlet series inversion, bypassing traditional zero-based methods.

Findings

Derived a power series for π(x) and J(x)

Established an inversion formula for Dirichlet series

Showed that zeros of the zeta function are not necessary for π(x)

Abstract

This paper discusses a few main topics in Number Theory, such as the M\"{o}bius function and its generalization, leading up to the derivation of neat power series for the prime counting function, $\pi(x)$, and the prime-power counting function, $J(x)$. Among its main findings, we can cite the extremely useful inversion formula for Dirichlet series (given $F_a(s)$, we know $a(n)$, which implies the Riemann hypothesis, and enabled the creation of a formula for $\pi(x)$ in the first place), and the realization that sums of divisors and the M\"{o}bius function are particular cases of a more general concept. From this result, one concludes that it's not necessary to resort to the zeros of the analytic continuation of the zeta function to obtain $\pi(x)$.