Editing Pafnuty Chebyshev (section)

== Mathematical contributions ==
[[File:Чебышёв, Пафнутий Львович.jpg|thumb|right|Pafnuty Chebyshev]]
Chebyshev is known for his work in the fields of [[probability]], [[statistics]], [[mechanics]], and [[number theory]]. The [[Chebyshev inequality]] states that if <math>X</math> is a [[random variable]] with [[standard deviation]] ''σ'' > 0, then the probability that the outcome of <math>X</math> is <math>d = k\sigma</math> or more away from its mean is at most <math>1/k^2 = \sigma^2/d^2</math>:

: <math>\Pr(|X - {\mathbf E}(X)| \ge d\ )\le \frac {\sigma^2}{d^2}.</math>

The Chebyshev inequality is used to prove the [[weak law of large numbers]].{{citation needed|date= October 2022}}

The [[Bertrand's postulate|Bertrand–Chebyshev theorem]] (1845, 1852) states that for any <math>n > 3</math>, there exists a [[prime numbers|prime number]] <math>p</math> such that <math>n < p < 2n</math>. This is a consequence of the Chebyshev inequalities for the number <math>\pi(n)</math> of [[prime numbers]] less than <math>n</math>:

: for <math> x </math> sufficiently large, <math> A\frac{x}{\log(x)}  < \pi(x) < \frac{6A}{5}\frac{x}{\log(x)}  \; \text{, with } \; A \approx 0.92129.</math><ref>{{cite journal | author=Tchebichef | title=Mémoire sur les nombres premiers | journal=Journal de Mathématiques Pures et Appliquées | date=1852 | issn=1776-3371 | pages=366–390 | url=https://eudml.org/doc/234762 | access-date=26 November 2024 | lang=fr}}  [https://nonagon.org/ExLibris/chebyshevs-memoire-nombres-premiers-section-1 English translation] by Mike Bertrand (November 5, 2020).</ref>

Fifty years later, in 1896, the celebrated [[prime number theorem]] was proved, independently, by [[Jacques Hadamard]]<ref name="Hadamard1896">{{Citation|last=Hadamard|first=Jacques|author-link=Jacques Hadamard|year=1896|title=Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques.|journal=Bulletin de la Société Mathématique de France|publisher=Société Mathématique de France|volume=24|pages=199–220|url=http://www.numdam.org/item/?id=BSMF_1896__24__199_1 |archive-url=https://web.archive.org/web/20240910153636/http://www.numdam.org/item/?id=BSMF_1896__24__199_1 |archive-date=2024-09-10 }}</ref> and [[Charles Jean de la Vallée Poussin]]:<ref name="de la Vallée Poussin1896">{{Citation|last=de la Vallée Poussin|first=Charles-Jean|author-link=Charles Jean de la Vallée Poussin|year=1896|title=Recherches analytiques sur la théorie des nombres premiers.|journal=Annales de la Société scientifique de Bruxelles|publisher=Imprimeur de l'Académie Royale de Belgique|volume=20 B; 21 B|pages=183-256, 281-352, 363-397; 351-368|url=http://sciences.amisbnf.org/fr/livre/recherches-analytiques-de-la-theorie-des-nombres-premiers}}</ref>

: <math>\lim_{x\to\infty}\frac{\;\pi(x)\log(x)\;}{x\;} = 1</math>

using ideas introduced by [[Bernhard Riemann]].

Chebyshev is also known for the [[Chebyshev polynomial]]s and the [[Chebyshev bias]] – the difference between the number of primes that are congruent to 3 (modulo 4) and 1 (modulo 4).<ref>{{cite journal | last1=Rubinstein | first1=Michael | last2=Sarnak | first2=Peter | title=Chebyshev's Bias | journal=Experimental Mathematics | volume=3 | issue=3 | date=1994 | issn=1058-6458 | doi=10.1080/10586458.1994.10504289 | pages=173–197}}</ref>

Chebyshev was the first person to think systematically in terms of [[random variables]] and their [[moment (mathematics)|moments]] and [[expected value|expectations]].<ref>{{cite journal|journal=Bulletin of the American Mathematical Society |series=New Series|volume=3|number=1|date=July 1980|last=Mackey|first=George|title=Harmonic analysis as the exploitation of symmetry-a historical survey|page=549|doi=10.1090/S0273-0979-1980-14783-7|doi-access=free|hdl=1911/63317|hdl-access=free}}</ref>