Reference for unconditional bounds of the sum $\sum_{n\leq x}\frac{\mu(n)}{n}$
问题内容
Could someone please provide information about the best possible known bounds of the sum $$A(x)=\sum_{n\leq x}\frac{\mu(n)}{n}?$$ Unconditionally, $A(x)=O(e^{-c\sqrt{\log x}})$ is known to me whose reference I need. Does there exist any better bound unconditionally?
Any help will be highly appreciated. Thanks.
回答 (1)
We have $A(x)=O(M(x)/x)$, see here. In particular, the known bounds for $M(x)=\sum_{n\le x}\mu(n)$ will give bounds for $A(x)$. A references for bounds on $M(x)$ is the paper by Schoenfeld, given in the above link; and a proper reference for both $M(x)$ and $A(x)$ is given in G.J.O. Jameson's The Prime Number Theorem, Thm. 5.1.9. page $186$ ($2003$).
The strongest unconditional bound uses the Korobov-Vinogradov zero-free region for the Riemann zeta function, and gives $$ A(x)=O(\exp(-c(\log (x))^{3/5}(\log (\log (x))^{-1/5})). $$ More details can be found at this MO-post.
Assuming RH yields $$\sum_{n\leq x} \frac{\mu(n)}{n} \ll \frac{\exp({(\log(x))^{1/2} (\log\log(x))^{14})}}{x^{1/2}}.$$