Editing Arithmetic function (section)

=== Menon's identity ===
In 1965 [[P Kesava Menon]] proved<ref>László Tóth, ''Menon's Identity and Arithmetical Sums ...'', eq. 1</ref>
<math display="block">\sum_{\stackrel{1\le k\le n}{ \gcd(k,n)=1}} \gcd(k-1,n)=\varphi(n)d(n).</math>

This has been generalized by a number of mathematicians. For example,
* B. Sury<ref>Tóth, eq. 5</ref> <math display="block">
\sum_{\stackrel{1\le k_1, k_2, \dots, k_s\le n}{ \gcd(k_1,n)=1}} \gcd(k_1-1,k_2,\dots,k_s,n)
= \varphi(n)\sigma_{s-1}(n).</math>
* N. Rao<ref>Tóth, eq. 3</ref> <math display="block">
\sum_{\stackrel{1\le k_1, k_2, \dots, k_s\le n}{ \gcd(k_1,k_2,\dots,k_s,n)=1}} \gcd(k_1-a_1,k_2-a_2,\dots,k_s-a_s,n)^s
=J_s(n)d(n), </math> where ''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''s''</sub> are integers, gcd(''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''s''</sub>, ''n'') = 1.
* [[László Fejes Tóth]]<ref>Tóth, eq. 35</ref> <math display="block">
\sum_{\stackrel{1\le k\le m}{ \gcd(k,m)=1}} \gcd(k^2-1,m_1)\gcd(k^2-1,m_2)
=\varphi(n)\sum_{\stackrel{d_1\mid m_1} {d_2\mid m_2}} \varphi(\gcd(d_1, d_2))2^{\omega(\operatorname{lcm}(d_1, d_2))},
</math> where ''m''<sub>1</sub> and ''m''<sub>2</sub> are odd, ''m'' = lcm(''m''<sub>1</sub>, ''m''<sub>2</sub>).

In fact, if ''f'' is any arithmetical function<ref>Tóth, eq. 2</ref><ref>Tóth states that Menon proved this for multiplicative ''f'' in 1965 and V. Sita Ramaiah for general ''f''.</ref>
<math display="block">\sum_{\stackrel{1\le k\le n}{ \gcd(k,n)=1}} f(\gcd(k-1,n))
=\varphi(n)\sum_{d\mid n}\frac{(\mu*f)(d)}{\varphi(d)},</math>
where <math>*</math> stands for Dirichlet convolution.