What is known about the asymptotic behavior of $$ f(x)=\sum_{p\le x}\frac1p-\log\log x-B_1? $$
Of course by Mertens we know that $f(x)=o(1),$ but has more been proved (in terms of $O$ or $\Omega_\pm$, say)? A reference would be great.
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Sign up to join this communityWhat is known about the asymptotic behavior of $$ f(x)=\sum_{p\le x}\frac1p-\log\log x-B_1? $$
Of course by Mertens we know that $f(x)=o(1),$ but has more been proved (in terms of $O$ or $\Omega_\pm$, say)? A reference would be great.
"Mertens' Proof of Mertens' Theorem" suggests that Mertens had an error term of $O\left(\frac1{\ln x}\right)$, though that's not tight; theorem 14 there offers an $O\left(\frac1{\ln^2x}\right)$ unconditional estimate and an $O(x^{-1/2}\ln x)$ estimate dependent on RH, with references to the original proofs.