Range criterion: Difference between revisions

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In [[number theory]], the '''Poussin proof''' is the proof of an identity related to the fractional part of a ratio.
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In 1838, [[Peter Gustav Lejeune Dirichlet]] proved an approximate formula for the average number of divisors of all the numbers from 1 to η:
 
:<math>\frac{\sum_{k=1}^\eta d(k)}{\eta} \approx \ln \eta + 2\gamma - 1,</math>
 
where ''d'' represents the [[divisor function]], and γ represents the [[Euler-Mascheroni constant]].
 
In 1898, [[Charles Jean de la Vallée-Poussin]] proved that if a large number η is divided by all the primes up to η, then the average fraction by which the quotient falls short of the next whole number is γ:
:<math>\frac{\sum_{p \leq \eta}\left \{ \frac{\eta}{p} \right \}}{\pi(\eta)} \approx1- \gamma,</math>
where {''x''} represents the [[fractional part]] of ''x'', and π represents the [[prime-counting function]].
For example, if we divide 29 by 2, we get 14.5, which falls short of 15 by 0.5.
 
==References==
*Dirichlet, G. L. "[http://gdz.sub.uni-goettingen.de/no_cache/en/dms/load/img/?IDDOC=268296 Sur l'usage des séries infinies dans la théorie des nombres]", ''Journal für die reine und angewandte Mathematik'' '''18''' (1838), pp.&nbsp;259–274. Cited in MathWorld article "Divisor Function" below.
*de la Vallée Poussin, C.-J. Untitled communication. ''Annales de la Societe Scientifique de Bruxelles'' '''22''' (1898), pp.&nbsp;84–90. Cited in MathWorld article "Euler-Mascheroni Constant" below.
 
==External links==
*{{MathWorld|urlname=DivisorFunction|title=Divisor Function}}
*{{MathWorld|urlname=Euler-MascheroniConstant|title=Euler-Mascheroni Constant}}
 
[[Category:Number theory]]
 
 
{{numtheory-stub}}

Latest revision as of 04:50, 12 March 2014

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