Take the tour ×  In Apostol's Analytic Number Theory, Apostol
defines $$A(x):= \sum_{n \leq x} \frac{\mu(n)}{n}$$ and proves
that $A(x)=o(1)$ implies the Prime Number Theorem, by showing
that $$\frac{M(x)}{x}=A(x)-\frac{1}{x}\int_1^x A(t)dt,$$ in
which $M(x):=\sum_{n \leq x} \mu(n)$ is the summatory function
for the Möbius function (Theorem 4.16). What are some known
error bounds for the function $A(x)$? In particular, do we have
$A(x)= o(1/\log x)$ as $x \to \infty$?  share|improve this
question Terence Tao has a rather interesting paper about this :
'A remark on partial sums involving the Mobius function'. See
too his blog. –  Raymond Manzoni Dec 7 '12 at 1:20 Schoenfeld's
paper, Marraki's one and Cohen's paper seem interesting too
(summatory function is much more studied than the $\zeta(1)$
formula). –  Raymond Manzoni Dec 7 '12 at 1:25 add comment  1
Answer  up vote 4 down vote accepted  I'll answer my own
question:  The Abel Summation formula gives
$$A(x)=\frac{M(x)}{x}+ \int_1^x \frac{M(u)}{u^2} du = \