- Poussin proof
In
number theory , the Poussin proof is the proof of an identity related to the fractional part of a ratio.In 1838, Dirichlet proved an approximate formula for the average number of divisors of all the numbers from 1 to η:
:frac{sum_{k=1}^eta d(k)}{eta} approx ln eta + 2gamma - 1,
where "d" represents the
divisor function , and γ represents theEuler-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::frac{sum_{p leq eta}left { frac{eta}{p} ight {pi(eta)} approx1- gamma,where {"x"} represents thefractional part of "x", and π represents theprime-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
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