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 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::$frac\{sum\_\{p\; leq\; eta\}left\; \{\; frac\{eta\}\{p\}\; ight\; \{pi(eta)\}\; approx1-\; gamma,$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