I am trying to solve a problem about the divisor function. Let us call $d(n)$ the classical divisor function, i.e. $d(n)=\sum\_{d|n}$ is the number of divisors of the integer $n$. It is well known that the sum of $d(n)$ over all positive integers from $n=1$ to $x$, when $x$ tends to infinity, is asymptotic to $$x \ln(x) + (2 \gamma-1) x + O(\sqrt{x})$$

I would like to calculate a similar asymptotic expression for the sum of $d(n)/n$, again calculated from $n=1$ to $x$ and for $x$ that tends to infinity. I have made some calculations and obtained the formula 
$$1/2 (\ln(x))^2 + 2 \gamma \ln (x) + O(1)$$ 
where gamma is the Euler-Mascheroni constant. I am interested in the constant term of the expression, which seems to be around $0.48$. I suspect that it could correspond to $\gamma^2 - 2\gamma\_1$, where $\gamma\_1$ is the first Stieltjes constant ($-0.072...$). Could someone confirm this to me? 

As an additional question, I would be very interested in obtaining similar asymptotic formulas, with explicitly given constant terms, for the same sum of $d(n)/n$ calculated over all odd integers from $1$ to $x$, and for that calculated over all even integers from $1$ to $x$. 
Many thanks.