Editing Arithmetic function (section)

== Completely multiplicative functions ==
=== ''λ''(''n'') – Liouville function ===
'''[[Liouville function|''λ''(''n'')]]''', the Liouville function, is  defined by
<math display="block">\lambda (n) = (-1)^{\Omega(n)}.</math>

=== ''χ''(''n'') – characters ===
All '''[[Dirichlet character]]s ''χ''(''n'')''' are completely multiplicative.  Two characters have special notations:

The '''principal character (mod ''n'')''' is denoted by ''χ''<sub>0</sub>(''a'') (or ''χ''<sub>1</sub>(''a'')). It is defined as
<math display="block"> \chi_0(a) = \begin{cases}
1 & \text{if } \gcd(a,n) = 1, \\
0 & \text{if } \gcd(a,n) \ne 1.
\end{cases} </math>

The '''quadratic character (mod ''n'')''' is denoted by the [[Jacobi symbol]] for odd ''n'' (it is not defined for even ''n''):
<math display="block">\left(\frac{a}{n}\right) = \left(\frac{a}{p_1}\right)^{a_1}\left(\frac{a}{p_2}\right)^{a_2}\cdots \left(\frac{a}{p_{\omega(n)}}\right)^{a_{\omega(n)}}.</math>

In this formula <math>(\tfrac{a}{p})</math> is the [[Legendre symbol]], defined for all integers ''a'' and all odd primes ''p'' by
<math display="block">
\left(\frac{a}{p}\right) = \begin{cases}
\;\;\,0 & \text{if } a \equiv 0 \pmod p,
\\+1 & \text{if }a \not\equiv 0\pmod p \text{ and for some integer }x, \;a\equiv x^2\pmod p
\\-1 & \text{if there is no such } x. \end{cases}</math>

Following the normal convention for the empty product, <math>\left(\frac{a}{1}\right) = 1.</math>