Return to Revisions

This Theorem 430, on page 355 of Hardy and Wright, that he "average order" of $\Omega(n)$ is $\log \log n.$ Then they point out, formula 22.10.2, that $$ \sum_{n \leq x} \; \Omega (n) \; = \; x \log \log x + B_2 x + o(x) $$ and say how to find the constant $$B_2 = B_1 + \sum_p \; \frac{1}{p(p-1)}. $$ Previously , $B_1$ was given as the constant in Merten's Theorem, first 427: $$ \sum_{p \leq x} \; \frac{1}{p} \; = \; \log \log x + B_1 + o(1), $$ then theorem 428, $$ B_1 = \gamma + \sum \left\{ \log \left( 1 - \frac{1}{p} \right) + \frac{1}{p} \right\}$$ so that $B_1 = 0.26149721\ldots$