Range criterion

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In number theory, 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 η:

${\frac {\sum _{k=1}^{\eta }d(k)}{\eta }}\approx \ln \eta +2\gamma -1,$

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:

${\frac {\sum _{p\leq \eta }\left\{{\frac {\eta }{p}}\right\}}{\pi (\eta )}}\approx 1-\gamma ,$

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. "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.

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