# Sum of Möbius Function

• Dec 28th 2010, 02:35 PM
chiph588@
Sum of Möbius Function
Let $\mu$ be the Möbius function and $k,n\in\mathbb{N}$, evaluate

$\displaystyle \sum_{d^k\mid n} \mu(d)$
• Dec 28th 2010, 02:51 PM
Bruno J.
We can factor this as $\prod_{p^{km}||n}(\mu(1)+\mu(p)+\dots+\mu(p^m))$ where $p^{km} || n$ means that $p^{km}|n$ but $p^{k(m+1)}\nmid n$. Clearly the product is $0$ unless $p^{0}||n$ for each $p$. Hence the sum is $1$ if $n$ is not divisible by a $k$-power (other than $1$), and $0$ otherwise.
• Dec 29th 2010, 12:26 PM
Bruno J.
So, is my solution accepted? :)
• Dec 29th 2010, 12:29 PM
chiph588@
Quote:

Originally Posted by Bruno J.
So, is my solution accepted? :)

Each term in the sum is $\mu(d)$, where $d=p_1^{a_1}\cdots p_j^{a_j}$ and $d^k = p_1^{ka_1}\cdots p_j^{ka_j}$ is such that $d^k | n$. In the product I wrote, the term $\mu(d)$ will be obtained as $\mu(p_1^{a_1})\cdots \mu(p_j^{a_j}) = \mu(d)$ once the product is expanded. Conversely, each term in the expansion of the product can be found in the sum.