Editing Analytic number theory (section)

===Chebyshev===
{{main|Pafnuty Chebyshev}}
In two papers from 1848 and 1850, the Russian mathematician [[Pafnuty Chebyshev|Pafnuty L'vovich Chebyshev]] attempted to prove the asymptotic law of distribution of prime numbers. His work is notable for the use of the zeta function ζ(''s'') (for real values of the argument "s", as are works of [[Leonhard Euler]], as early as 1737) predating Riemann's celebrated memoir of 1859, and he succeeded in proving a slightly weaker form of the asymptotic law, namely, that if the limit of π(''x'')/(''x''/ln(''x'')) as ''x'' goes to infinity exists at all, then it is necessarily equal to one.<ref>{{cite journal |author=N. Costa Pereira |jstor=2322510 |title=A Short Proof of Chebyshev's Theorem |journal=American Mathematical Monthly|date=August–September 1985|pages=494–495|volume=92|doi=10.2307/2322510|issue=7}}</ref> He was able to prove unconditionally that this ratio is bounded above and below by two explicitly given constants near to 1 for all ''x''.<ref>{{cite journal |author=M. Nair |jstor=2320934 |title=On Chebyshev-Type Inequalities for Primes |journal=American Mathematical Monthly |date=February 1982 |pages=126–129 |volume=89  |doi=10.2307/2320934 |issue=2}}</ref> Although Chebyshev's paper did not prove the Prime Number Theorem, his estimates for π(''x'') were strong enough for him to prove [[Bertrand's postulate]] that there exists a prime number between ''n'' and 2''n'' for any integer ''n''&nbsp;≥&nbsp;2.