The Möbius Function and the Möbius Transform

Abstract

After Euler’s totient function, the Möbius function (named after the Möbius of strip fame) is one of the most important tools of number theory. It allows us to invert certain number-theoretic relations. In a sense, if we liken summation over divisors to integration, then taking the Möbius function is like differentiating. It is defined as follows:

$$ \mu (n): = \left\{ {\begin{array}{*{20}{c}} {1 {\kern 1pt} for {\kern 1pt} n = 1} \\ {0{\kern 1pt} if{\kern 1pt} n {\kern 1pt} is {\kern 1pt} divisible {\kern 1pt} by {\kern 1pt} a {\kern 1pt} square} \\ {{{( - 1)}^K}{\kern 1pt} if{\kern 1pt} n{\kern 1pt} {\kern 1pt} is{\kern 1pt} the{\kern 1pt} product{\kern 1pt} of{\kern 1pt} k{\kern 1pt} distinct{\kern 1pt} {\kern 1pt} primes} \end{array}} \right. $$

(20.1)

Thus, µ(n)≠ 0 only for 1 and squarefree integers. The Möbius function is multiplicative:$\mu (mn) = \mu (m)\mu (n){\kern 1pt} if(m,n) = 1$ Also,

$$ \mu (mn) = 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} if(m,n) >1 $$

(20.2)

because then mn is not squarefree.