 Turán–Kubilius inequality

For an additive arithmetic function $f$, $N^{-1}{\sum }_{n\le N}|f(n)-A_f(N){|}^2\ll B_f(N{)}^2$, where $A_f$ is the mean of an additive arithmetic function and $B_f(N{)}^2={\sum }_{p^k\le N}|f(p^k){|}^2/p^k$. The implied constant is absolute.

The Turán–Kubilius inequality is a result in number theory and probabilistic number theory, often related to the distribution of prime numbers. It provides a bound on the probability that certain events, often concerning the sums of random variables, will occur.