An Exact Formula for the Prime Counting Function

Jose R. Sousa

An Exact Formula for the Prime Counting Function

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 a neat power series for the prime counting function, $\pi(x)$. Among its main findings, we can cite the extremely useful inversion formula for Dirichlet series (given $F_a(s)$, we know $a(n)$, which may provide evidence for 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. One of its conclusions is that it's unnecessary to resort to the zeros of the analytic continuation of the zeta function to obtain $\pi(x)$.

Comments: 21 Pages.

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[v1] 2021-11-25 21:44:32

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