Poussin proof explained

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:

n
\sumd(k)
k=1
n

lnn+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:

\sum\left\{
n
p
\right\
p\leqn
} \approx1- \gamma,where 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.

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