Editing Prime number theorem (section)

== Statement ==
[[File:Prime number theorem ratio convergence.svg|thumb|300px|Graph showing ratio of the prime-counting function {{math|''π''(''x'')}} to two of its approximations, {{math|''x'' / log ''x''}} and {{math|Li(''x'')}}. As {{mvar|x}} increases (note {{mvar|x}} axis is logarithmic), both ratios tend towards 1. The ratio for {{math|''x'' / log ''x''}} converges from above very slowly, while the ratio for {{math|Li(''x'')}} converges more quickly from below.]]
[[File:Prime number theorem absolute error.svg|thumb|300px|Log–log plot showing absolute error of {{math|''x'' / log ''x''}} and {{math|Li(''x'')}}, two approximations to the prime-counting function {{math|''π''(''x'')}}. Unlike the ratio, the difference between {{math|''π''(''x'')}} and {{math|''x'' / log ''x''}} increases without bound as {{mvar|x}} increases. On the other hand, {{math|Li(''x'') − ''π''(''x'')}} switches sign infinitely many times.]]
<!--[[Image:PrimeNumberTheorem.svg|thumb|right|250px|Graph comparing {{math|''π''(''x'')}} (red), {{math|''x'' / log ''x''}} (green) and {{math|Li(''x'')}} (blue)]] -->

Let {{math|''π''(''x'')}} be the [[prime-counting function]] defined to be the number of primes less than or equal to {{mvar|x}}, for any real number&nbsp;{{mvar|x}}. For example, {{math|''π''(10) {{=}} 4}} because there are four prime numbers (2, 3, 5 and 7) less than or equal to 10. The prime number theorem then states that {{math|''x'' / log ''x''}} is a good approximation to {{math|''π''(''x'')}} (where log here means the natural logarithm), in the sense that the [[limit of a function|limit]] of the ''quotient'' of the two functions {{math|''π''(''x'')}} and {{math|''x'' / log ''x''}} as {{mvar|x}} increases without bound is 1:
: <math>\lim_{x\to\infty}\frac{\;\pi(x)\;}{\;\left[ \frac{x}{\log(x)}\right]\;} = 1,</math>
known as the '''asymptotic law of distribution of prime numbers'''. Using [[asymptotic notation]] this result can be restated as
: <math>\pi(x)\sim \frac{x}{\log x}.</math>

This notation (and the theorem) does ''not'' say anything about the limit of the ''difference'' of the two functions as {{mvar|x}} increases without bound. Instead, the theorem states that {{math|''x'' / log ''x''}} approximates {{math|''π''(''x'')}} in the sense that the [[relative error]] of this approximation approaches 0 as {{mvar|x}} increases without bound.

The prime number theorem is equivalent to the statement that the {{mvar|n}}th prime number {{mvar|p<sub>n</sub>}} satisfies
: <math>p_n \sim n\log(n),</math>
the asymptotic notation meaning, again, that the relative error of this approximation approaches 0 as {{mvar|n}} increases without bound. For example, the {{val|2|e=17}}th prime number is {{val|8512677386048191063}},<ref>{{cite web|title=Prime Curios!: 8512677386048191063|url=http://primes.utm.edu/curios/cpage/24149.html|work=Prime Curios!|publisher=University of Tennessee at Martin|date=2011-10-09}}</ref> and ({{val|2|e=17}})log({{val|2|e=17}}) rounds to {{val|7967418752291744388}}, a relative error of about 6.4%.

On the other hand, the following asymptotic relations are logically equivalent:<ref name=Apostol76>{{cite book |last1=Apostol |first1=Tom M. |author-link=Tom M. Apostol |title=Introduction to Analytic Number Theory |series=Undergraduate Texts in Mathematics |year=1976 |publisher=Springer |isbn=978-1-4757-5579-4 |doi=10.1007/978-1-4757-5579-4 |edition=1 |url=https://link.springer.com/book/10.1007/978-1-4757-5579-4#about}}</ref>{{rp|80–82}}
: <math>\begin{align}
    \lim_{x\rightarrow \infty}\frac{\pi(x)\log x}{x}&=1,\text{ and}\\
    \lim_{x\rightarrow \infty}\frac{\pi(x)\log \pi(x)}{x}\,&=1.
\end{align}
</math>

As outlined [[#Proof sketch|below]], the prime number theorem is also equivalent to
: <math>\lim_{x\to\infty} \frac{\vartheta (x)}x = \lim_{x\to\infty} \frac{\psi(x)}x=1,</math>
where {{mvar|{{not a typo|ϑ}}}} and {{mvar|ψ}} are [[Chebyshev function|the first and the second Chebyshev functions]] respectively, and to
: <math>\lim_{x \to \infty} \frac{M(x)}{x}=0,</math>{{r|Apostol76|p=92–94}}
where <math>M(x)=\sum_{n \leq x} \mu(n)</math> is the [[Mertens function]].