Poussin proof

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 n:

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

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 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:

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

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.