Editing Prime number theorem (section)

== Non-asymptotic bounds on the prime-counting function ==
{{main|Prime-counting function#Inequalities}}
The prime number theorem is an ''asymptotic'' result. It gives an [[Effective results in number theory|ineffective]] bound on {{math|''π''(''x'')}} as a direct consequence of the definition of the limit: for all {{math|''ε'' > 0}}, there is an {{mvar|S}} such that for all {{math|''x'' > ''S''}},
: <math> (1-\varepsilon)\frac {x}{\log x} \; < \; \pi(x) \; < \; (1+\varepsilon)\frac {x}{\log x} \; .</math>

However, better bounds on {{math|''π''(''x'')}} are known, for instance [[Pierre Dusart]]'s
: <math> \frac{x}{\log x}\left(1+\frac{1}{\log x}\right) \; < \; \pi(x) \; < \; \frac{x}{\log x}\left(1+\frac{1}{\log x}+\frac{2.51}{(\log x)^2}\right) \; .</math>
The first inequality holds for all {{math|''x'' ≥ 599}} and the second one for {{math|''x'' ≥ 355991}}.<ref>{{cite thesis |last=Dusart |first=Pierre |author-link=Pierre Dusart |date=26 May 1998 |title=Autour de la fonction qui compte le nombre de nombres premiers |trans-title=About the prime-counting function |degree=Ph.D. |place=Limoges, France |publisher=l'Université de Limoges |department=département de Mathématiques |url=https://www.unilim.fr/pages_perso/pierre.dusart/Documents/T1998_01.pdf |lang=fr}}</ref>

The proof by de la Vallée Poussin implies the following bound: For every {{math|''ε'' > 0}}, there is an {{mvar|S}} such that for all {{math|''x'' > ''S''}},
: <math>\frac {x}{\log x - (1 - \varepsilon)} \; < \; \pi(x) \; < \; \frac {x}{\log x - (1+\varepsilon)} \; .</math>

The value {{math|''ε'' {{=}} 3}} gives a weak but sometimes useful bound for {{math|''x'' ≥ 55}}:<ref name="rosser">{{cite journal |first=Barkley |last=Rosser |author-link=J. Barkley Rosser |year=1941 |title=Explicit bounds for some functions of prime numbers |journal=[[American Journal of Mathematics]] |volume=63 |issue=1 |pages=211–232 |doi=10.2307/2371291 |jstor=2371291 |mr=0003018}}</ref>
: <math> \frac {x}{\log x + 2} \; < \; \pi(x) \; < \; \frac {x}{\log x - 4} \; .</math>

In Pierre Dusart's thesis there are stronger versions of this type of inequality that are valid for larger {{mvar|x}}. Later in 2010, Dusart proved:<ref>{{cite arXiv |last=Dusart |first=Pierre |author-link=Pierre Dusart |date=2 February 2010 |title=Estimates of some functions over primes, without {{abbr|R.H.|Riemann hypothesis}} |eprint=1002.0442 |class=math.NT}}</ref>
: <math>\begin{align}
\frac {x}{\log x - 1} \; &< \; \pi(x)                   &&\text{ for } x \ge 5393 \;, \text{ and }\\
\pi(x)                   &< \; \frac {x} {\log x - 1.1} &&\text{ for } x \ge 60184 \; .
\end{align}</math>

Note that the first of these obsoletes the {{math|''ε'' > 0}} condition on the lower bound.