Editing Prime-counting function (section)

{{short description|Function representing the number of primes less than or equal to a given number}}
{{Redirect|Π(x)|the variant of the gamma function|Gamma function#Pi function}} 
{{log(x)}}
{{Duplication|dupe=Prime number theorem|discuss=Talk:Prime number theorem#Too much duplication in Prime number theorem and Prime-counting function|date=December 2024}}
[[Image:PrimePi.svg|thumb|right|400px|The values of {{math|''π''(''n'')}} for the first 60 positive integers]]

In [[mathematics]], the '''prime-counting function''' is the [[Function (mathematics)|function]] counting the number of [[prime number]]s less than or equal to some [[real number]] {{mvar|x}}.<ref name="Bach">{{cite book |first=Eric |last=Bach |author2=Shallit, Jeffrey |year=1996 |title=Algorithmic Number Theory |publisher=MIT Press |isbn=0-262-02405-5 |pages=volume 1 page 234 section 8.8 |no-pp=true}}</ref><ref name="mathworld_pcf">{{MathWorld |title=Prime Counting Function |urlname=PrimeCountingFunction}}</ref> It is denoted by {{math|''π''(''x'')}} (unrelated to the [[pi|number {{pi}}]]).

A symmetric variant seen sometimes is {{math|''π''<sub>0</sub>(''x'')}}, which is equal to {{math|''π''(''x'') − {{frac|1|2}}}} if {{mvar|x}} is exactly a prime number, and equal to {{math|''π''(''x'')}} otherwise.  That is, the number of prime numbers less than {{mvar|x}}, plus half if {{mvar|x}} equals a prime.