Poussin proof

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:

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

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 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:

:$\backslash frac\{\backslash sum\_\{p\; \backslash leq\; n\}\backslash left\; \backslash \{\; \backslash frac\{n\}\{p\}\; \backslash right\; \backslash \}\}\{\backslash pi(n)\}\; \backslash approx1-\; \backslash gamma,$

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.