Summing a multiplicative function

 Published on Sunday, 21st October 2018, 07:00 am; Solved by 221;
Difficulty rating: 70%

Problem 639

A multiplicative function $f(x)$ is a function over positive integers satisfying $f(1)=1$ and $f(a b)=f(a) f(b)$ for any two coprime positive integers $a$ and $b$.

For integer $k$ let $f_k(n)$ be a multiplicative function additionally satisfying $f_k(p^e)=p^k$ for any prime $p$ and any integer $e>0$.
For example, $f_1(2)=2$, $f_1(4)=2$, $f_1(18)=6$ and $f_2(18)=36$.

Let $\displaystyle S_k(n)=\sum_{i=1}^{n} f_k(i)$. For example, $S_1(10)=41$, $S_1(100)=3512$, $S_2(100)=208090$, $S_1(10000)=35252550$ and $\displaystyle \sum_{k=1}^{3} S_k(10^{8}) \equiv 338787512 \pmod{ 1\,000\,000\,007}$.

Find $\displaystyle \sum_{k=1}^{50} S_k(10^{12}) \bmod 1\,000\,000\,007$.