I want to solve this problem:
Let the number of divisors = d(n) (for example, d(6)=4 because number 6 has 4 divisors, {1, 2, 3, 6}), I want to calculate d(1)+d(2)+d(3)+...+d(n).
But I can't calculate for large n like 10^20 or 10^30 (I think the algorithm calculting in a few seconds if n is as large as 10^30 doesn't exist), so I decided to find the algorithm that calculates approximately.
I found that the answer is near by n log n (the base of log is e=2.71828...)
But in case of n = 10^17 the error is about 0.4%.
It's a little accurate, but I think that there are more accurate algorithm.
Is there any more accurate algorithm?
The Encyclopedia of Mathematics credits the estimate
n log n + (2γ - 1)n + O(√n)
to Dirichlet (1849). γ is the Euler–Mascheroni constant. You can just drop the O(√n) error term.