"Weighted" sum of divisor function
Hi, I am trying to solve a problem about the divisor function. Let us call d(n) the classical divisor function, i.e. 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 infinite, 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 infinite. I have made some calculations and believe that this formula may be 1/2 (Ln(x))^2 + 2 gamma Ln (x) +O(1), where the last term of the expression, that is to say the intercept, approximately tends to 0.96. Since I am not sure of this formula, could someone help me? If the formula is correct, I would particularly be interested to know what that intercept exactly is.
As an additional question, I would be very interested in obtaining similar asymptotic formulas for the same sum of d(n)/n calculated over all odd integers from 1 to x, and calculated over all even integers from 1 to x.