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Problem 608

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Divisor Sums

Let $D(m,n)=\displaystyle\sum_{d|m}\sum_{k=1}^n\sigma_{\small 0}(kd)$ where $d$ runs through all divisors of $m$ and $\sigma_{\small 0}(n)$ is the number of divisors of $n$.
You are given $D(3!,10^2)=3398$ and $D(4!,10^6)=268882292$.

Find $D(200!,10^{12}) \text{ mod } (10^9 + 7)$.

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因数和

记$D(m,n)=\displaystyle\sum_{d|m}\sum_{k=1}^n\sigma_{\small 0}(kd)$，其中$d$取遍$m$的所有因数，而$\sigma_{\small 0}(n)$表示$n$的因数数目。
已知$D(3!,10^2)=3398$以及$D(4!,10^6)=268882292$。

求$D(200!,10^{12}) \text{ mod } (10^9 + 7)$。

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