Turán normal-order theorem for distinct prime divisors
= Turán normal-order theorem for distinct prime divisors
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The <prime omega function> has normal order $\log\log n$. The elementary second-moment estimate
$$
\sum_{n\leq x}\bigl(\omega(n)-\log\log x\bigr)^2
\ll x\log\log x
$$
already proves concentration on every scale larger than $\sqrt{\log\log x}$.