Let $d(k)$ be the sum of all divisors of $k$.
We define the function $S(N) = \sum_{i=1}^N \sum_{j=1}^Nd(i \cdot j)$.
For example, $S(3) = d(1) + d(2) + d(3) + d(2) + d(4) + d(6) + d(3) + d(6) + d(9) = 59$.
You are given that $S(10^3) = 563576517282$ and $S(10^5) \bmod 10^9 = 215766508$.
Find $S(10^{11}) \bmod 10^9$.
Solution
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