Poussin proof

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The Poussin proof is a proof of a formula in the number theory area of pure mathematics.

In 1838, Dirichlet proved an approximate formula for the average number of divisors of all the numbers from 1 to η:

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

where d represents the divisor function, and γ represents the Euler-Mascheroni constant.
In 1898, Charles-Joseph 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. For example, if we divide 29 by 2, we get 14.5, which falls short of 15 by 0.5. In symbols, this would be:

$\frac{\sum_{p \leq \eta}\left \{ \frac{\eta}{p} \right \}}{\pi(\eta)} \approx1- \gamma,$