Editing Prime number theorem (section)

== Prime-counting function in terms of the logarithmic integral ==
In a handwritten note on a reprint of his 1838 paper "{{lang|fr|Sur l'usage des séries infinies dans la théorie des nombres}}", which he mailed to Gauss, Dirichlet conjectured (under a slightly different form appealing to a series rather than an integral) that an even better approximation to {{math|''π''(''x'')}} is given by the [[logarithmic integral function|offset logarithmic integral]] function {{math|Li(''x'')}}, defined by
: <math> \operatorname{Li}(x) = \int_2^x \frac{dt}{\log t} = \operatorname{li}(x) - \operatorname{li}(2). </math>

Indeed, this integral is strongly suggestive of the notion that the "density" of primes around {{mvar|t}} should be {{math|1 / log ''t''}}. This function is related to the logarithm by the [[asymptotic expansion]]
: <math> \operatorname{Li}(x) \sim \frac{x}{\log x} \sum_{k=0}^\infty \frac{k!}{(\log x)^k} = \frac{x}{\log x} + \frac{x}{(\log x)^2} + \frac{2x}{(\log x)^3} + \cdots </math>

So, the prime number theorem can also be written as {{math|''π''(''x'') ~ Li(''x'')}}. In fact, in another paper<ref name="de la Vallée Poussin1899">{{citation|last=de la Vallée Poussin|first=Charles-Jean|author-link=Charles Jean de la Vallée Poussin|year=1899|title=Sur la fonction ζ(s) de Riemann et le nombre des nombres premiers inférieurs a une limite donnée.|journal=Mémoires couronnés de l'Académie de Belgique|publisher=Imprimeur de l'Académie Royale de Belgique|volume=59|pages=1–74|url={{Google Books|_O0GAAAAYAAJ|Sur la fonction ζ(s) de Riemann et le nombre des nombres premiers inférieurs a une limite donnée.|plainurl=yes}}}}</ref> in 1899 de la Vallée Poussin proved that
: <math> \pi(x) = \operatorname{Li} (x) + O \left(x e^{-a\sqrt{\log x}}\right) \quad\text{as } x \to \infty</math>
for some positive constant {{mvar|a}}, where {{math|''O''(...)}} is the [[big O notation|big {{mvar|O}} notation]]. This has been improved to
: <math>\pi(x) = \operatorname{li} (x) + O \left(x \exp \left( -\frac{A(\log x)^\frac35}{(\log \log x)^\frac15} \right) \right)</math> where <math>A = 0.2098</math>.<ref name="Ford">{{cite journal |author = Kevin Ford |author-link = Kevin Ford (mathematician) |title=Vinogradov's Integral and Bounds for the Riemann Zeta Function |journal=Proc. London Math. Soc. |date=2002 |volume=85 |issue = 3 |pages=565–633 |url=https://faculty.math.illinois.edu/~ford/wwwpapers/zetabd.pdf |doi=10.1112/S0024611502013655 |arxiv = 1910.08209 |s2cid = 121144007 }}</ref>

In 2016, [[Timothy Trudgian]] proved an explicit upper bound for the difference between <math>\pi(x)</math> and <math>\operatorname{li}(x)</math>:
: <math>\big| \pi(x) - \operatorname{li}(x) \big| \le 0.2795 \frac{x}{(\log x)^{3/4}} \exp \left( -\sqrt{ \frac{\log x}{6.455} } \right)</math>
for <math>x \ge 229</math>.<ref>{{cite journal |author = Timothy Trudgian | author-link=Timothy Trudgian | date = February 2016 |title = Updating the error term in the prime number theorem |journal = Ramanujan Journal |volume = 39 |issue = 2  |pages=225–234 |doi = 10.1007/s11139-014-9656-6 |arxiv = 1401.2689 |s2cid = 11013503 }}</ref>

The connection between the Riemann zeta function and {{math|''π''(''x'')}} is one reason the [[Riemann hypothesis]] has considerable importance in number theory: if established, it would yield a far better estimate of the error involved in the prime number theorem than is available today. More specifically, [[Helge von Koch]] showed in 1901<ref>{{cite journal |first=Helge |last=von Koch |year=1901 |title=Sur la distribution des nombres premiers |journal=[[Acta Mathematica]] |volume=24 |issue=1 |pages=159–182 |doi=10.1007/BF02403071 |lang=fr |trans-title=On the distribution of prime numbers|mr=1554926 |s2cid=119914826 |url=https://zenodo.org/record/2347595|doi-access=free }}</ref> that if the Riemann hypothesis is true, the error term in the above relation can be improved to
: <math> \pi(x) = \operatorname{Li} (x) + O\left(\sqrt x \log x\right) </math>
(this last estimate is in fact equivalent to the Riemann hypothesis). The constant involved in the big {{mvar|O}} notation was estimated in 1976 by [[Lowell Schoenfeld]],<ref>{{cite journal |last=Schoenfeld |first=Lowell |title=Sharper Bounds for the Chebyshev Functions {{math|''{{not a typo|ϑ}}''(''x'')}} and {{math|''ψ''(''x'')}}. II |journal=Mathematics of Computation |volume=30 |issue=134 |year=1976 |pages=337–360 |doi=10.2307/2005976 |jstor=2005976 | mr=0457374 }}</ref> assuming the Riemann hypothesis:
: <math>\big|\pi(x) - \operatorname{li}(x)\big| < \frac{\sqrt x \log x}{8\pi}</math>
for all {{math|''x'' ≥ 2657}}. He also derived a similar bound for the [[Chebyshev function|Chebyshev prime-counting function]] {{mvar|ψ}}:
: <math>\big|\psi(x) - x\big| < \frac{\sqrt x (\log x)^2 }{8\pi}</math>
for all {{math|''x'' ≥ 73.2}}&nbsp;. This latter bound has been shown to express a variance to mean [[power law]] (when regarded as a random function over the integers) and {{sfrac| {{mvar|f}} }}&nbsp;[[pink noise|noise]] and to also correspond to the [[Tweedie distribution|Tweedie compound Poisson distribution]]. (The Tweedie distributions represent a family of [[scale invariant]] distributions that serve as foci of convergence for a generalization of the [[central limit theorem]].<ref>{{cite journal |last1=Jørgensen |first1=Bent |last2=Martínez |first2=José Raúl |last3=Tsao |first3=Min |year=1994 |title=Asymptotic behaviour of the variance function |journal=Scandinavian Journal of Statistics |volume=21 |issue=3 |pages=223–243 |mr=1292637 |jstor=4616314 }}</ref>) A lower bound is also derived by [[John Edensor Littlewood|J. E. Littlewood]], assuming the Riemann hypothesis:<ref name="Littlewood1914">{{citation |first=J.E. |last= Littlewood |author-link=John Edensor Littlewood |year=1914 |title=Sur la distribution des nombres premiers |journal=[[Comptes Rendus]] |volume=158 |pages= 1869–1872 | jfm=45.0305.01}}</ref><ref>{{cite journal |first1=G. H. |last1=Hardy |author-link1=G. H. Hardy |first2=J. E. |last2=Littlewood |author-link2=John Edensor Littlewood |year=1916 |title=Contributions to the theory of the Riemann zeta-function and the theory of the distribution of primes |journal=[[Acta Mathematica]] |volume=41 |pages=119–196 |doi=10.1007/BF02422942 |url=https://link.springer.com/article/10.1007/BF02422942}}</ref><ref>
{{cite book
 |last1=Davenport |first1=Harold |author1-link=Harold Davenport
 |last2=Montgomery |first2=Hugh L. |author2-link=Hugh Montgomery (mathematician)
 |year=2000
 |title=Multiplicative Number Theory
 |edition=revised 3rd
 |series=Graduate Texts in Mathematics
 |volume=74
 |publisher=[[Springer Publishing|Springer]]
 |isbn=978-0-387-95097-6
}}</ref>
: <math>\big|\pi(x) - \operatorname{li}(x)\big| = \Omega \left(\sqrt x\frac{\log\log\log x}{\log x} \right)</math>

The [[Logarithmic integral function|logarithmic integral]] {{math|li(''x'')}} is larger than {{math|''π''(''x'')}} for "small" values of {{mvar|x}}. This is because it is (in some sense) counting not primes, but prime powers, where a power {{mvar|p{{sup|n}}}} of a prime {{mvar|p}} is counted as {{sfrac|1| {{mvar|n}} }} of a prime. This suggests that {{math|li(''x'')}} should usually be larger than {{math|''π''(''x'')}} by roughly <math>\ \tfrac{1}{2} \operatorname{li}(\sqrt{x})\ ,</math> and in particular should always be larger than {{math|''π''(''x'')}}. However, in 1914, Littlewood proved that <math>\ \pi(x) - \operatorname{li}(x)\ </math> changes sign infinitely often.<ref name="Littlewood1914"/>  The first value of {{mvar|x}} where {{math|''π''(''x'')}} exceeds {{math|li(''x'')}} is probably around {{math|''x'' ~ {{10^|316}} }}; see the article on [[Skewes' number]] for more details. (On the other hand, the [[offset logarithmic integral]] {{math|Li(''x'')}} is smaller than {{math|''π''(''x'')}} already for {{math|''x'' {{=}} 2}}; indeed, {{math|Li(2) {{=}} 0}}, while {{math|''π''(2) {{=}} 1}}.)