Vrftjkry

It is very well known and not difficult to prove that

$displaystylesum_{substack{0<nleq X\ \(n,q)=1}}frac{1}{n}=left(log(X)+gamma+sum_{p|q}frac{log(p)}{p-1}right)frac{phi(q)}{q}+Oleft(frac{2^{omega(q)}}{X}right)$.

Moreover, from the above expression, we obtain that

$displaystylesum_{substack{0<nleq X\ \(n,q)=1}}frac{1}{n} sim left(log(X)+gamma+sum_{p|q}frac{log(p)}{p-1}right)frac{phi(q)}{q}$

On the other hand, for $q=1$, we can obtain an optimal and explicit error term, for example, thanks to the Euler-Maclaurin summation formula, given by $O^*left(frac{1}{2X}right)$ (we can obtain many lower order terms by using this, too). Inspired by this, the above error term $Oleft(frac{2^{omega(q)}}{q}right)$ looks like improvable, at least explicitly (i.e. with better constants, in $O^*$ notation, say), if not in order.

The question is which would be the best known bound, with optimal constants, and why, for the following expressions

$star: displaystyleleft|sum_{substack{0<nleq X\ \(n,q)=1}}frac{1}{n}-left(log(X)+gamma+sum_{p|q}frac{log(p)}{p-1}right)frac{phi(q)}{q}right|$,

and in general, for $X_1neq 0$,

$starstar: displaystyleleft|sum_{substack{X_1<nleq X_2\ \(n,q)=1}}frac{1}{n}-frac{phi(q)}{q}logleft(frac{X_2}{X_1}right)right|$?

And related, is there an explicit asymptotic formula involving many lower order terms?

I haven't found any good explanation for this matter in the literature, so your help would be really appreciated. Thanks!