M\"obius functionMoebiusFunction2001-10-16 09:08:532005-07-26 22:42:43Definitionnumber theory% this is the default PlanetMath preamble. as your knowledge
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% define commands hereThe {\em M\"obius function} of number theory is the function $\mu:\mathbb{Z}^+\to\{-1,0,1\}$ defined by
\[
\mu (n) =
\begin{cases}
1, &\text{if $n=1$}\\
0, &\text{if $p^2 | n$ for some prime $p$} \\
(-1)^r, &\text{if $n = p_1 p_2 \cdots p_r$, where the $p_i$ are distinct primes.}
\end{cases}
\]
In other words, $\mu (n) = 0$ if $n$ is not a square-free integer, while $\mu (n) = (-1)^r$ if $n$ is square-free with $r$ prime factors. The function $\mu$ is a multiplicative function, and obeys the identity
\[
\sum_{d | n} \mu(d) =
\begin{cases}
1 & \text{if $n = 1$}\\
0 & \text{if $n > 1$}
\end{cases}
\]
where $d$ runs through the positive divisors of $n$.