除数函数 的公式

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除數函數 $\sigma _{x}(n)$ 为 $n$ 的正约数的 $x$ 次幂之和. 则
$$\sigma _{x}(n)=\sum _{\mu =1}^{n}\mu ^{x-1}\sum _{\nu =1}^{\mu }\cos {\frac {2\pi \nu n}{\mu }}$$

证明

只需证明\begin{equation}\sum _{\nu =1}^{\mu }\cos {\frac {2\pi \nu n}{\mu }}=\begin{cases}\mu&\mu\mid n\\
0&\mu\nmid n
\end{cases}\end{equation}当 $\mu\mid n$ 时, 每一项都是1; 当 $\mu\nmid n$ 时,
$$2\sum _{\nu =1}^{\mu }\cos {\frac {2\pi \nu n}{\mu }}=\sum _{\nu =1}^{\mu }\exp \left(\nu{\frac {2\pi i n}{\mu }}\right)+\exp\left(-\nu{\frac {2\pi i n}{\mu }}\right)$$使用等比数列求和公式、然后$\exp\left((μ+1)\frac{2\pi i n}μ\right)-\exp\left(\frac{2\pi i n}μ\right)=0$✅

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Dirichlet series involving the divisor function: [ Hardy & Wright (2008), pp. 326–328, §17.5. ]
\[\sum _{n=1}^{\infty }{\frac {\sigma _{a}(n)}{n^{s}}}=\zeta (s)\zeta (s-a).\]
Ramanujan identity [ Hardy & Wright (2008), pp. 334–337, §17.8. ] is a special case of the Rankin–Selberg convolution.
\[\sum _{{n=1}}^{{\infty }}{\frac {\sigma _{a}(n)\sigma _{b}(n)}{n^{s}}}={\frac {\zeta (s)\zeta (s-a)\zeta (s-b)\zeta (s-a-b)}{\zeta (2s-a-b)}}.\]