Problem 639

Summing a multiplicative function

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

For integer $k$ let $f_k(n)$ be a multiplicative function additionally satisfying $f_k(e^p)=p^k$ 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 $S_k(n)=\sum_{i=1}^n f_k(i)$. For example, $S_1(10)=41$, $S_1(100)=3512$, $S_2(100)=208090$, $S_1(10000)=35252550$ and $\sum_{k=1}^3 S_k(10^8)\equiv 338787512 (\mod 1\ 000\ 000\ 007)$.

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

积性函数求和

若定义在正整数上的函数$f(x)$满足$f(1)=1$，且对任意两个互质正整数$a$和$b$有$f(ab)=f(a)f(b)$，则称$f(x)$为积性函数。

对于正整数$k$，积性函数$f_k(n)$进一步满足，对于任意质数$p$和任意整数$e>0$有$f_k(e^p)=p^k$。
例如，$f_1(2)=2$，$f_1(4)=2$，$f_1(18)=6$，$f_2(18)=36$。

记$S_k(n)=\sum_{i=1}^n f_k(i)$。例如，$S_1(10)=41$，$S_1(100)=3512$，$S_2(100)=208090$，$S_1(10000)=35252550$，$\sum_{k=1}^3 S_k(10^8)\equiv 338787512 (\mod 1\ 000\ 000\ 007)$。

求$\sum_{k=1}^{50}S_k(10^{12})\mod 1\ 000\ 000\ 007$。