Exact Form of Prime-Counting Function

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Theorem

Let:

$\displaystyle \map \Pi x = \map \Li x - \sum_\rho \map \Li {x^\rho} - \map \ln 2 + \int_x^\infty \frac {\d t} {t \paren {t^2 - 1} \, \map \ln t}$

where:

$\map \Li x$ is the offset logarithmic integral
the sum $\displaystyle \sum_\rho$ is taken over all $0 < \rho \in \R$ such that the zeta function $\map \zeta {\alpha + i \rho} = 0$ for some $\alpha \in \R$.


Then the prime-counting function is precisely:

$\displaystyle \map \pi x = \sum_{n \mathop = 1}^\infty \paren {\frac {\map \mu n} n \map \Pi {x^{1/n} } }$

where $\map \mu n$ denotes the Möbius function.


Proof