Editing Prime-counting function (section)

===Exact form===

For {{math|''x'' > 1}} let {{math|''π''<sub>0</sub>(''x'') {{=}} ''π''(''x'') − {{sfrac|1|2}}}} when {{mvar|x}} is a prime number, and {{math|''π''<sub>0</sub>(''x'') {{=}} ''π''(''x'')}} otherwise. [[Bernhard Riemann]], in his work ''[[On the Number of Primes Less Than a Given Magnitude]]'', proved that {{math|''π''<sub>0</sub>(''x'')}} is equal to<ref>{{Cite web|url=http://ism.uqam.ca/~ism/pdf/Hutama-scientific%20report.pdf|title=Implementation of Riemann's Explicit Formula for Rational and Gaussian Primes in Sage|last=Hutama|first=Daniel|date=2017|website=Institut des sciences mathématiques}}</ref>[[File:Riemann Explicit Formula.gif|thumb|Riemann's explicit formula using the first 200 non-trivial zeros of the zeta function|402x402px]]
<math display="block">\pi_0(x) = \operatorname{R}(x) - \sum_{\rho}\operatorname{R}(x^\rho),</math>
where
<math display=block>\operatorname{R}(x) = \sum_{n=1}^{\infty} \frac{\mu(n)}{n} \operatorname{li}\left(x^{1/n}\right),</math>
{{math|''μ''(''n'')}} is the [[Möbius function]], {{math|li(''x'')}} is the [[logarithmic integral function]], {{mvar|ρ}} indexes every zero of the Riemann zeta function, and {{math|li(''x''<sup>{{sfrac|''ρ''|''n''}}</sup>)}} is not evaluated with a [[branch cut]] but instead considered as {{math|Ei({{sfrac|''ρ''|''n''}} log ''x'')}} where {{math|Ei(''x'')}} is the [[exponential integral]]. If the trivial zeros are collected and the sum is taken ''only'' over the non-trivial zeros {{mvar|ρ}} of the Riemann zeta function, then {{math|''π''<sub>0</sub>(''x'')}} may be approximated by<ref name="RieselGohl">{{Cite journal | author1-link=Hans Riesel | last1=Riesel | first1=Hans | last2=Göhl | first2=Gunnar | title=Some calculations related to Riemann's prime number formula | doi=10.2307/2004630 | mr=0277489  | year=1970 | journal=[[Mathematics of Computation]] | issn=0025-5718 | volume=24 | issue=112 | pages=969–983 | jstor=2004630 | publisher=American Mathematical Society |url=https://www.ams.org/journals/mcom/1970-24-112/S0025-5718-1970-0277489-3/S0025-5718-1970-0277489-3.pdf }}</ref>
<math display=block>\pi_0(x) \approx \operatorname{R}(x) - \sum_{\rho}\operatorname{R}\left(x^\rho\right) - \frac{1}{\log x} + \frac{1}{\pi} \arctan{\frac{\pi}{\log x}} .</math>

The [[Riemann hypothesis]] suggests that every such non-trivial zero lies along {{math|1=Re(''s'') = {{sfrac|1|2}}}}.