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In [[number theory]], a branch of [[mathematics]], the '''Poussin proof''' is the proof of an identity related to the [[fractional part]] of a [[ratio]].

In 1838, [[Peter Gustav Lejeune Dirichlet]] proved an approximate formula for the average number of divisors of all the numbers from 1 to n:

:<math>\frac{\sum_{k=1}^n d(k)}{n} \approx \ln n + 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 n is divided by all the primes up to n, then the average fraction by which the quotient falls short of the next whole number is γ:
:<math>\frac{\sum_{p \leq n}\left \{ \frac{n}{p} \right \}}{\pi(n)} \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. 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. 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}}

Poussin proof - Revision history

imported>David Eppstein: more context