Editing Arithmetic function (section)

=== Dirichlet convolutions ===
: <math>
\sum_{\delta\mid n}\mu(\delta)=
\sum_{\delta\mid n}\lambda\left(\frac{n}{\delta}\right)|\mu(\delta)|=
\begin{cases}
1 & \text{if } n=1\\
0 & \text{if } n\ne1
\end{cases}
</math> &nbsp; &nbsp; where ''λ'' is the Liouville function.<ref>Hardy & Wright, Thm. 263</ref>
: <math>\sum_{\delta\mid n}\varphi(\delta) = n.</math> &nbsp; &nbsp; &nbsp;<ref>Hardy & Wright, Thm. 63</ref>
:: <math>\varphi(n)
=\sum_{\delta\mid n}\mu\left(\frac{n}{\delta}\right)\delta
=n\sum_{\delta\mid n}\frac{\mu(\delta)}{\delta}.
</math> &nbsp; &nbsp; &nbsp;  Möbius inversion
: <math>\sum_{d \mid n } J_k(d) = n^k.</math> &nbsp; &nbsp; &nbsp;<ref>see references at [[Jordan's totient function]]</ref>
:: <math>
J_k(n)
=\sum_{\delta\mid n}\mu\left(\frac{n}{\delta}\right)\delta^k
=n^k\sum_{\delta\mid n}\frac{\mu(\delta)}{\delta^k}.
</math> &nbsp; &nbsp; &nbsp;  Möbius inversion
: <math>\sum_{\delta\mid n}\delta^sJ_r(\delta)J_s\left(\frac{n}{\delta}\right) = J_{r+s}(n)</math> &nbsp; &nbsp; &nbsp;<ref>Holden et al. in external links The formula is Gegenbauer's</ref>
: <math>\sum_{\delta\mid n}\varphi(\delta)d\left(\frac{n}{\delta}\right) = \sigma(n).</math> &nbsp; &nbsp; &nbsp;<ref>Hardy & Wright, Thm. 288–290</ref><ref>Dineva in external links, prop. 4</ref>
: <math>\sum_{\delta\mid n}|\mu(\delta)| = 2^{\omega(n)}.</math> &nbsp; &nbsp; &nbsp;<ref>Hardy & Wright, Thm. 264</ref>
:: <math>|\mu(n)|=\sum_{\delta\mid n}\mu\left(\frac{n}{\delta}\right)2^{\omega(\delta)}.</math> &nbsp; &nbsp; &nbsp;  Möbius inversion
: <math>\sum_{\delta\mid n}2^{\omega(\delta)}=d(n^2).</math> &nbsp; &nbsp; &nbsp;
:: <math>2^{\omega(n)}=\sum_{\delta\mid n}\mu\left(\frac{n}{\delta}\right)d(\delta^2).</math> &nbsp; &nbsp; &nbsp;  Möbius inversion
: <math>\sum_{\delta\mid n}d(\delta^2)=d^2(n).</math> &nbsp; &nbsp; &nbsp;
:: <math>d(n^2)=\sum_{\delta\mid n}\mu\left(\frac{n}{\delta}\right)d^2(\delta).</math> &nbsp; &nbsp; &nbsp;  Möbius inversion
: <math>\sum_{\delta\mid n}d\left(\frac{n}{\delta}\right)2^{\omega(\delta)}=d^2(n).</math> &nbsp; &nbsp; &nbsp;
: <math>\sum_{\delta\mid n}\lambda(\delta)=\begin{cases}
&1\text{ if } n \text{ is a square }\\
&0\text{ if } n \text{ is not  square.}
\end{cases}</math>  &nbsp; &nbsp;  where λ is the [[Liouville function]].
: <math>\sum_{\delta\mid n}\Lambda(\delta) = \log n.</math> &nbsp; &nbsp; &nbsp;<ref>Hardy & Wright, Thm. 296</ref>
:: <math>\Lambda(n)=\sum_{\delta\mid n}\mu\left(\frac{n}{\delta}\right)\log(\delta).</math> &nbsp; &nbsp; &nbsp;  Möbius inversion