Editing Prime-counting function (section)

==Growth rate==
{{main|Prime number theorem}}
Of great interest in [[number theory]] is the [[Asymptotic analysis|growth rate]] of the prime-counting function.<ref name="Caldwell">{{cite web | publisher=Chris K. Caldwell | title=How many primes are there? | url=http://primes.utm.edu/howmany.shtml | access-date=2008-12-02 | archive-date=2012-10-15 | archive-url=https://web.archive.org/web/20121015002415/http://primes.utm.edu/howmany.shtml | url-status=dead }}</ref><ref name="Dickson">{{cite book |author-link=L. E. Dickson| first=Leonard Eugene | last=Dickson | year=2005 | title=History of the Theory of Numbers, Vol. I: Divisibility and Primality | publisher=Dover Publications | isbn=0-486-44232-2}}</ref> It was [[conjecture]]d in the end of the 18th century by [[Carl Friedrich Gauss|Gauss]] and by [[Adrien-Marie Legendre|Legendre]] to be approximately
<math display=block> \frac{x}{\log x} </math>
where {{math|log}} is the [[natural logarithm]], in the sense that
<math display=block>\lim_{x\rightarrow\infty} \frac{\pi(x)}{x/\log x}=1. </math>
This statement is the [[prime number theorem]]. An equivalent statement is
<math display=block>\lim_{x\rightarrow\infty}\frac{\pi(x)}{\operatorname{li}(x)}=1</math>
where {{math|li}} is the [[logarithmic integral]] function. The prime number theorem was first proved in 1896 by [[Jacques Hadamard]] and by [[Charles Jean de la Vallée-Poussin|Charles de la Vallée Poussin]] independently, using properties of the [[Riemann zeta function]] introduced by [[Bernhard Riemann|Riemann]] in 1859. Proofs of the prime number theorem not using the zeta function or [[complex analysis]] were found around 1948 by [[Atle Selberg]] and by [[Paul Erdős]] (for the most part independently).<ref name="Ireland">{{cite book | first=Kenneth | last=Ireland |author2=Rosen, Michael | year=1998 | title=A Classical Introduction to Modern Number Theory  | edition=Second | publisher=Springer | isbn=0-387-97329-X }}</ref>

===More precise estimates===
In 1899, [[Charles Jean de la Vallée Poussin|de la Vallée Poussin]] proved that
<ref>See also Theorem 23 of {{cite book |author = A. E. Ingham |author-link = Albert Ingham |title = The Distribution of Prime Numbers |date=2000 |publisher = Cambridge University Press |isbn=0-521-39789-8}}</ref>
<math display=block>\pi(x) = \operatorname{li} (x) + O \left(x e^{-a\sqrt{\log x}}\right) \quad\text{as } x \to \infty</math>
for some positive constant {{mvar|a}}. Here, {{math|''O''(...)}} is the [[big O notation|big {{mvar|O}} notation]].

More precise estimates of {{math|''π''(''x'')}} are now known. For example, in 2002, [[Kevin Ford (mathematician)|Kevin Ford]] proved that<ref name="Ford">{{cite journal |author = Kevin Ford |title=Vinogradov's Integral and Bounds for the Riemann Zeta Function |journal=Proc. London Math. Soc. |date=November 2002 |volume=85 |issue=3 |pages=565–633 |url=https://faculty.math.illinois.edu/~ford/wwwpapers/zetabd.pdf |doi=10.1112/S0024611502013655 |arxiv=1910.08209 |s2cid=121144007 }}</ref>
<math display=block>\pi(x) = \operatorname{li} (x) + O \left(x \exp \left( -0.2098(\log x)^{3/5} (\log \log x)^{-1/5} \right) \right).</math>

Mossinghoff and [[Timothy Trudgian|Trudgian]] proved<ref>{{cite journal | first1 = Michael J. | last1 = Mossinghoff  | first2 = Timothy S. | last2 = Trudgian | author2-link=Timothy Trudgian| title = Nonnegative trigonometric polynomials and a zero-free region for the Riemann zeta-function | journal = J. Number Theory | volume = 157 | year = 2015 | pages = 329–349 | arxiv = 1410.3926 | doi = 10.1016/J.JNT.2015.05.010| s2cid = 117968965 }}</ref> an explicit upper bound for the difference between {{math|''π''(''x'')}} and {{math|li(''x'')}}:
<math display=block>\bigl| \pi(x) - \operatorname{li}(x) \bigr| \le 0.2593 \frac{x}{(\log x)^{3/4}} \exp \left( -\sqrt{ \frac{\log x}{6.315} } \right) \quad \text{for } x \ge 229.</math>

For values of {{mvar|x}} that are not unreasonably large, {{math|li(''x'')}} is greater than {{math|''π''(''x'')}}. However, {{math|''π''(''x'') − li(''x'')}} is known to change sign infinitely many times. For a discussion of this, see [[Skewes' number]].

===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}}}}.