Editing Prime-counting function (section)

===Riemann's prime-power counting function===
Riemann's prime-power counting function is usually denoted as {{math|Π<sub>0</sub>(''x'')}} or {{math|''J''<sub>0</sub>(''x'')}}. It has jumps of {{math|{{sfrac|1|''n''}}}} at prime powers {{mvar|p<sup>n</sup>}} and it takes a value halfway between the two sides at the discontinuities of {{math|''π''(''x'')}}. That added detail is used because the function may then be defined by an inverse [[Mellin transform]].

Formally, we may define {{math|Π<sub>0</sub>(''x'')}} by
:<math>\Pi_0(x) = \frac{1}{2} \left( \sum_{p^n < x} \frac{1}{n} + \sum_{p^n \le x} \frac{1}{n} \right)\ </math>
where the variable {{mvar|p}} in each sum ranges over all primes within the specified limits.

We may also write
:<math>\ \Pi_0(x) = \sum_{n=2}^x \frac{\Lambda(n)}{\log n} - \frac{\Lambda(x)}{2\log x} = \sum_{n=1}^\infty \frac 1 n \pi_0\left(x^{1/n}\right)</math>
where {{math|Λ}} is the [[von Mangoldt function]] and

:<math>\pi_0(x) = \lim_{\varepsilon \to 0} \frac{\pi(x-\varepsilon) + \pi(x+\varepsilon)}{2}.</math>

The [[Möbius inversion formula]] then gives
:<math>\pi_0(x) = \sum_{n=1}^\infty \frac{\mu(n)}{n}\ \Pi_0\left(x^{1/n}\right),</math>
where {{math|''μ''(''n'')}} is the [[Möbius function]].

Knowing the relationship between the logarithm of the [[Riemann zeta function]] and the [[von Mangoldt function]] {{math|Λ}}, and using the [[Perron formula]] we have
:<math>\log \zeta(s) = s \int_0^\infty \Pi_0(x) x^{-s-1}\, \mathrm{d}x</math>