Number Theory


An Exact Formula for the Prime Counting Function

Authors: Jose R. Sousa

This article 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 inversion theorem for Dirichlet series (given $F_a(s)$, we can tell what its associated function, $a(n)$, is), which 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. Another conclusion we draw is that it's unnecessary to resort to the zeros of the analytic continuation of the zeta function to obtain $\pi(x)$.

Comments: 16 Pages. I think this finding may have interesting applications in the study of the Riemann Hypothesis

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Submission history

[v1] 2019-07-29 22:01:51

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