Editing Prime number theorem (section)

== Approximations for the ''n''th prime number ==
As a consequence of the prime number theorem, one gets an asymptotic expression for the {{mvar|n}}th prime number, denoted by {{math|''p''<sub>''n''</sub>}}:
: <math>p_n \sim n \log n.</math><ref>{{cite web |title=Why is p<sub>n</sub> ∼ n ln(n)? |url=https://math.stackexchange.com/questions/1413167/why-is-p-n-sim-n-lnn |access-date=2024-10-11 |website=Mathematics Stack Exchange |language=en}}</ref>
A better approximation is by [[Ernesto Cesàro|Cesàro]] (1894):<ref>{{cite journal|author-link=Ernesto Cesàro|first=Ernesto|last=Cesàro|year=1894|title=Sur une formule empirique de M. Pervouchine|journal=Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences|volume=119|pages=848–849|url=http://gallica.bnf.fr/ark:/12148/bpt6k30752|language=fr}}</ref>
: <math> p_n = n B_2(\log n),\text{ where}</math>
: <math> B_2(x) = x + \log x - 1 + \frac{\log x - 2}{x} - \frac{(\log x)^2 - 6 \log x + 11}{2x^2} + o\left(\frac 1{x^2}\right).</math>

Again considering the {{val|2|e=17}}th prime number {{val|8512677386048191063}}, assuming the [[Little-o notation|trailing error term]] is zero gives an estimate of {{val|8512681315554715386}}; the first 5 digits match and relative error is about 0.46 [[parts per million]].

[[Michele Cipolla|Cipolla]] (1902)<ref>{{cite journal
 |title=La determinazione assintotica dell'n<sup>imo</sup> numero primo
 |trans-title=The asymptotic determination of the n<sup>th</sup> prime number
 |lang=it
 |first=Michele |last=Cipolla |author-link=Michele Cipolla
 |journal=Matematiche Napoli
 |series=8 |volume=3 |year=1902 |pages=132–166
}}</ref><ref name=Toulisse13>{{cite journal
 |title= The n-th prime asymptotically
 |first1=Juan |last1=Arias de Reyna
 |first2=Jérémy |last2=Toulisse
 |journal=Journal de théorie des nombres de Bordeaux
 |lang=en
 |volume=25 |issue=3 |year=2013 |pages=521–555
 |doi=10.5802/jtnb.847 |doi-access=free
 |mr=3179675 |zbl=1298.11093 |arxiv=1203.5413
}}</ref> showed that these are the leading terms of an infinite series which may be truncated at arbitrary degree, with
: <math>B_k(x) = x + \log x - 1 - \sum_{i=1}^k (-1)^i \frac{P_i(\log x)}{ix^i} + O\left(\frac{(\log x)^{k+1}}{x^{k+1}}\right),</math>
where each {{math|''P<sub>i</sub>''}} is a degree-{{mvar|i}} monic polynomial.  ({{math|1=''P''<sub>1</sub>(''y'') = ''y'' − 2}}, {{math|1=''P''<sub>2</sub>(''y'') = ''y''<sup>2</sup> − 6''y'' + 11}}, {{math|1=''P''<sub>3</sub>(''y'') = ''y''<sup>3</sup> − {{sfrac|21|2}}''y''<sup>2</sup> + 42''y'' + {{sfrac|131|2}}}}, and so on.<ref name=Toulisse13/>)

[[Rosser's theorem]]<ref name="rosser" /> states that 
: <math>p_n > n \log n.</math> 
Dusart (1999).<ref name=Dusart99>{{cite journal|author-link=Pierre Dusart|last=Dusart|first=Pierre|title=The {{mvar|k}}th prime is greater than {{math|''k''(log ''k'' + log log ''k'' − 1)}} for {{math|''k'' ≥ 2}}|journal=[[Mathematics of Computation]]|volume=68|issue=225|year=1999|pages=411–415|mr=1620223|doi=10.1090/S0025-5718-99-01037-6|doi-access=free}}</ref> found tighter bounds using the form of the Cesàro/Cipolla approximations but varying the lowest-order constant term.  {{math|''B<sub>k</sub>''(''x''; ''C'')}} is the same function as above, but with the lowest-order constant term replaced by a parameter {{mvar|C}}:<!--Note that this notation is created for Wikipedia to keep the equations manageably short, and is not from any source.  B is for "bound"; change it if you like.-->
: <math>\begin{align}
p_n \; &> \; n B_0(\log n; 1) && \text{for } n \ge 2,\text{ and}\\
p_n \; &< \; n B_0(\log n; 0.9484) && \text{for } n \ge 39017,\text{ where}\\
B_0(x;C) \;  &= \; x + \log x - C.\\
p_n \; &> \; n B_1(\log n; 2.25) && \text{for } n \ge 2,\text{ and}\\
p_n \; &< \; n B_1(\log n; 1.8) && \text{for } n \ge 27076,\text{ where}\\
B_1(x;C) \;  &= \; x + \log x - 1 + \frac{\log x - C}{x}.
\end{align}</math>
The upper bounds can be extended to smaller {{mvar|n}} by loosening the parameter.  For example, {{math|''p<sub>n</sub>'' < ''n B''<sub>1</sub>(log ''n''; 0.5)}} for all {{math|''n'' &geq; 20}}.<ref name=Axler19>{{cite journal
 |journal=Journal of Integer Sequences
 |volume=22 |year=2019 |article-number=19.4.2
 |title=New Estimates for the {{mvar|n}}th Prime Number
 |first=Christian |last=Axler
 |arxiv=1706.03651
 |url=https://cs.uwaterloo.ca/journals/JIS/VOL22/Axler/axler17.html
}}</ref>

Axler (2019)<ref name=Axler19/> extended this to higher order, showing:
: <math>\begin{align}
p_n \; &> \; n B_2(\log n;11.321) \quad \text{for } n \ge 2, \text{ and }\\
p_n \; &< \; n B_2(\log n;10.667) \quad \text{for } n \ge 46\,254\,381,\text{ where}\\
B_2(x;C) \;  &= \; x + \log x - 1 + \frac{\log x - 2}{x} - \frac{(\log x)^2 - 6\log x + C}{2x^2}.
\end{align}</math>
Again, the bound on {{mvar|n}} may be decreased by loosening the parameter.  For example, {{math|''p<sub>n</sub>'' < ''n B''<sub>2</sub>(log ''n''; 0)}} for {{math|n &geq; 3468}}.