New answers tagged mobius-function
3
votes
Finding a "Closed Form" Expression for $\sum_{d|n}\mu(n/d)\tau(d)$
We consider some basics of arithmetical functions, i.e. functions from $f:\mathbb{N}\to\mathbb{C}$ and show this way the claim. We want to show
\begin{align*}
\sum_{d|n}\mu(n/d)\tau(d)=1\qquad\qquad n\...
- 111k
2
votes
Accepted
Minimal number of generators of the monoid generated by roots of the unity.
I have edited as I got a stronger result:
Note that $-1\in M_n$ for all $n$, since
$$\zeta_n^{n-1}+\zeta_n^{n-2}+...+1=0$$
$$\implies \zeta_n^{n-1}+\zeta_n^{n-2}+...+\zeta_n=-1\in M_n$$
Then ...
- 9,250
1
vote
Unicity of decomposition for the monoid generated by roots of the unity.
This monoid is contained in the abelian group underlying the ring of integers $\mathbb{Z}[\zeta_n]$ of the cyclotomic field $\mathbb{Q}[\zeta_n]$. The field has $\mathbb{Q}$-dimension $\phi(n)$ (where ...
- 9,250
1
vote
Unicity of decomposition for the monoid generated by roots of the unity.
First note that $U_n=\{\zeta_n^m: m=0,\ldots,{n-1}\}$, where $\zeta_n$ is any primitive $n$-th root of unity, and so
$$M_n=\left\{\sum_{i=1}^nk_i\zeta_n^i:\ k_0,\ldots,k_{n-1}\in\Bbb{N}\right\}.$$
...
- 65.6k
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