Exercise 9.1.14

Question

\def\imp{\Rightarrow}
\def\gcro{\mbox{[\hspace{-.15em}[}}% intervalles d'entiers 
\def\dcro{\mbox{]\hspace{-.15em}]}}

ewcommand{\be} {\begin{enumerate}}

ewcommand{\ee} {\end{enumerate}}

ewcommand{\deb}{\begin{eqnarray*}}

ewcommand{\fin}{\end{eqnarray*}}

ewcommand{\ssi} {si et seulement si }

ewcommand{\D}{\mathrm{d}}

ewcommand{\Q}{\mathbb{Q}}

ewcommand{\Z}{\mathbb{Z}}

ewcommand{\N}{\mathbb{N}}

ewcommand{\R}{\mathbb{R}}

ewcommand{\C}{\mathbb{C}}

ewcommand{\F}{\mathbb{F}}

ewcommand{\U}{\mathbb{U}}

ewcommand{e}{\,\mathrm{Re}\,}

ewcommand{\im}{\,\mathrm{Im}\,}

ewcommand{\ord}{\mathrm{ord}}

ewcommand{\Gal}{\mathrm{Gal}}

ewcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}}

The M"obius function is defined for integers $n\geq 1$ by
$$
\mu(n) = 
\left\{
\begin{array}{ll}
  1&    \mathrm{if} \ n=1,  \
  (-1)^s,& \mathrm{if}\ n =p_1\cdots p_s\  \mathrm{for}\  \mathrm{distinct}\ \mathrm{primes}\ p_1, \ldots,p_s    \
  0,    &   \mathrm{otherwise}
\end{array}
ight.
$$
Prove that $\sum_{d \mid n} \mu\left(\frac{n}{d}ight) = 0$ when $n>1$.

Richard Ganaye · Accepted Answer

\def\imp{\Rightarrow}
\def\gcro{\mbox{[\hspace{-.15em}[}}% intervalles d'entiers 
\def\dcro{\mbox{]\hspace{-.15em}]}}

ewcommand{\be} {\begin{enumerate}}

ewcommand{\ee} {\end{enumerate}}

ewcommand{\deb}{\begin{eqnarray*}}

ewcommand{\fin}{\end{eqnarray*}}

ewcommand{\ssi} {si et seulement si }

ewcommand{\D}{\mathrm{d}}

ewcommand{\Q}{\mathbb{Q}}

ewcommand{\Z}{\mathbb{Z}}

ewcommand{\N}{\mathbb{N}}

ewcommand{\R}{\mathbb{R}}

ewcommand{\C}{\mathbb{C}}

ewcommand{\F}{\mathbb{F}}

ewcommand{\U}{\mathbb{U}}

ewcommand{e}{\,\mathrm{Re}\,}

ewcommand{\im}{\,\mathrm{Im}\,}

ewcommand{\ord}{\mathrm{ord}}

ewcommand{\Gal}{\mathrm{Gal}}

ewcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}}

\begin{proof}
Suppose $n>1$.  Write $n =p_1^{\alpha_1} p_2^{\alpha_2}\cdots p_k^{\alpha_k}$ its decomposition in prime factors. The factors $d$ of $n$ such that $\mu(d) 
e 0$ are the integers $d = p_1^{\beta_1} p_2^{\beta_2}\cdots p_k^{\beta_k}$ where $\beta_i = 0,1$. If exactly $r$  exponents $\beta_i$ are non zero, then $\mu(d) = (-1)^r$, and there are  $\binom{k}{r}$ such integers $d$.
 
Therefore $$\sum_{d \vert n} \mu(d) = \sum_{r=0}^{k} (-1)^r \binom{k}{r} = (1-1)^k = 0 $$ (since $k 
ot = 0$)

Conclusion: if $n \geq 1$,
 $$\sum_{d \mid n} \mu\left(\frac{n}{d}ight) = \sum_{d \vert n} \mu(d)= 
 \left\{
\begin{array}{ll}
  0,&    \mathrm{if} \ n>1,  \
  1,   &   \mathrm{if}\ n=1.
\end{array}
ight.
$$
\end{proof}

Exercise 9.1.14

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Exercise 9.1.14

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