Editing Prime-counting function (section)

==Inequalities==
[[Srinivasa Ramanujan|Ramanujan]]<ref>{{Cite book|url=https://books.google.com/books?id=QoMHCAAAQBAJ|title=Ramanujan's Notebooks, Part IV|last=Berndt|first=Bruce C.|date=2012-12-06|pages=112–113|publisher=Springer Science & Business Media|isbn=9781461269328|language=en}}</ref> proved that the inequality
:<math>\pi(x)^2 < \frac{ex}{\log x} \pi\left( \frac{x}{e} \right)</math>
holds for all sufficiently large values of {{mvar|x}}.

Here are some useful inequalities for {{math|''π''(''x'')}}.

:<math> \frac x {\log x} < \pi(x) < 1.25506 \frac x {\log x} \quad \text{for }x \ge 17.</math>

The left inequality holds for {{math|''x'' ≥ 17}} and the right inequality holds for {{math|''x'' > 1}}. The constant {{#expr:30*ln(113)/113 round 5}} is {{math|30{{sfrac|log 113|113}}}} to 5 decimal places, as {{math|''π''(''x'') {{sfrac|log ''x''|''x''}}}} has its maximum value at {{math|1=''x'' = ''p''<sub>30</sub> = 113}}.<ref name=Rosser1962>{{Cite journal | author-link = J. Barkley Rosser | last1 = Rosser | first1 = J. Barkley | last2 = Schoenfeld | first2 = Lowell | title = Approximate formulas for some functions of prime numbers | journal = [[Illinois J. Math.]] | year = 1962 | volume = 6 | pages = 64–94 | doi = 10.1215/ijm/1255631807 | zbl = 0122.05001 | issn = 0019-2082 | url = https://projecteuclid.org/euclid.ijm/1255631807 | doi-access = free }}</ref>

[[Pierre Dusart]] proved in 2010:<ref name = "Dusart2010">{{cite arXiv  |last = Dusart |first = Pierre |author-link = Pierre Dusart |eprint=1002.0442v1 |title = Estimates of Some Functions Over Primes without R.H. |class = math.NT |date = 2 Feb 2010 }}</ref>

:<math> \frac {x} {\log x - 1} < \pi(x) < \frac {x} {\log x - 1.1}\quad \text{for }x \ge 5393 \text{ and }x \ge 60184,\text{ respectively.}</math>

More recently, Dusart has proved<ref>{{cite journal |last = Dusart |first = Pierre |author-link = Pierre Dusart |title = Explicit estimates of some functions over primes |journal = Ramanujan Journal |volume = 45 |issue = 1 |pages=225–234 |date = January 2018 |doi = 10.1007/s11139-016-9839-4|s2cid = 125120533 }}</ref>
(Theorem 5.1) that
:<math>\frac{x}{\log x} \left( 1 + \frac{1}{\log x} + \frac{2}{\log^2 x} \right) \le \pi(x) \le \frac{x}{\log x} \left( 1 + \frac{1}{\log x} + \frac{2}{\log^2 x} + \frac{7.59}{\log^3 x} \right),</math>
for {{math|''x'' ≥ 88789}} and  {{math|''x'' > 1}}, respectively.

Going in the other direction, an approximation for the {{mvar|n}}th  prime, {{mvar|p<sub>n</sub>}}, is
:<math>p_n = n \left(\log n + \log\log n - 1 + \frac {\log\log n - 2}{\log n} + O\left( \frac {(\log\log n)^2} {(\log n)^2}\right)\right).</math>

Here are some inequalities for the {{mvar|n}}th prime. The lower bound is due to Dusart (1999)<ref>{{cite journal
| author-link=Pierre Dusart
| last       = Dusart
| first      = Pierre
| date       = January 1999
| title      = The ''k<sup>th</sup>'' prime is greater than ''k''(ln ''k'' + ln ln ''k'' − 1) for ''k'' ≥ 2
| journal    = Mathematics of Computation
| volume     = 68
| issue      = 225
| pages      = 411–415
| doi        = 10.1090/S0025-5718-99-01037-6
| doi-access = free
| bibcode    = 1999MaCom..68..411D
| url        = https://www.ams.org/mcom/1999-68-225/S0025-5718-99-01037-6/S0025-5718-99-01037-6.pdf
}}</ref> and the upper bound to Rosser (1941).<ref>{{cite journal
| first      = Barkley | last = Rosser | author-link = J. Barkley Rosser
| date       = January 1941
| title      = Explicit bounds for some functions of prime numbers
| jstor      = 2371291
| journal    = American Journal of Mathematics
| volume     = 63
| issue      = 1
| pages      = 211–232
| doi        = 10.2307/2371291
}}</ref>

:<math> n (\log n + \log\log n - 1) < p_n < n (\log n + \log\log n)\quad \text{for } n \ge 6.</math>

The left inequality holds for {{math|''n'' ≥ 2}} and the right inequality holds for {{math|''n'' ≥ 6}}.  A variant form sometimes seen substitutes <math>\log n +\log\log n = \log(n \log n).</math>  An even simpler lower bound is<ref name=Rosser62>{{cite journal
| title      = Approximate formulas for some functions of prime numbers
| first1     = J. Barkley | last1 = Rosser | author1-link = J. Barkley Rosser
| first2     = Lowell | last2 = Schoenfeld | author2-link = Lowell Schoenfeld
| journal    = Illinois Journal of Mathematics
| volume     = 6
| issue      = 1
| pages      = 64–94
| date       = March 1962
| doi        = 10.1215/ijm/1255631807
}}</ref>
:<math>n \log n < p_n,</math>
which holds for all {{math|''n'' ≥ 1}}, but the lower bound above is tighter for {{math|''n'' > ''e<sup>e</sup>'' &asymp;{{#expr:exp(exp(1)) round 3}}}}.

In 2010 Dusart proved<ref name = "Dusart2010" /> (Propositions 6.7 and 6.6) that
:<math> n \left( \log n + \log \log n - 1 + \frac{\log \log n - 2.1}{\log n} \right) \le p_n \le n \left( \log n + \log \log n - 1 + \frac{\log \log n - 2}{\log n} \right),</math>
for {{math|''n'' ≥ 3}} and {{math|''n'' ≥ 688383}}, respectively.

In 2024, Axler<ref>{{cite journal
| title      = New estimates for the ''n''th prime number
| first      = Christian | last = Axler
| journal    = Journal of Integer Sequences
| volume     = 19
| issue      = 4
| article-number = 2
| date       = 2019
| orig-date  = 23 Mar 2017
| arxiv      = 1706.03651
| url        = https://cs.uwaterloo.ca/journals/JIS/VOL22/Axler/axler17.html
}}</ref> further tightened this (equations 1.12 and 1.13) using bounds of the form
:<math> f(n,g(w)) = n \left( \log n + \log\log n - 1 + \frac{\log\log n - 2}{\log n} - \frac{g(\log\log n)}{2\log^2 n} \right)</math>
proving that
:<math> f(n, w^2 - 6w + 11.321) \le p_n \le f(n, w^2 - 6w)</math>
for {{math|''n'' ≥ 2}} and {{math|''n'' ≥ 3468}}, respectively.
The lower bound may also be simplified to {{math|''f''(''n'', ''w''<sup>2</sup>)}} without altering its validity.  The upper bound may be tightened to {{math|''f''(''n'', ''w''<sup>2</sup> − 6''w'' + 10.667)}} if {{math|''n'' ≥ 46254381}}.

There are additional bounds of varying complexity.<ref>{{cite web
| title      = Bounds for ''n''-th prime
| url        = https://math.stackexchange.com/questions/1270814/bounds-for-n-th-prime
| date       = 31 December 2015
| website    = Mathematics StackExchange
}}</ref><ref>{{cite journal
| title      = New Estimates for Some Functions Defined Over Primes 
| first      = Christian | last = Axler
| journal    = Integers
| volume     = 18
| article-number = A52
| doi        = 10.5281/zenodo.10677755
| doi-access = free
| date       = 2018
| orig-date  = 23 Mar 2017
| arxiv      = 1703.08032
| url        = https://math.colgate.edu/~integers/s52/s52.pdf
}}</ref><ref>{{cite journal
| title      = Effective Estimates for Some Functions Defined over Primes 
| first      = Christian | last = Axler
| journal    = Integers
| volume     = 24
| article-number = A34
| doi        = 10.5281/zenodo.10677755
| doi-access = free
| date       = 2024
| orig-date  = 11 Mar 2022
| arxiv      = 2203.05917 
| url        = https://math.colgate.edu/~integers/y34/y34.pdf
}}</ref>