Exercise 2.12

Question

\def\imp{\Rightarrow}
\def\gcro{\mbox{[\hspace{-.15em}[}}
\def\dcro{\mbox{]\hspace{-.15em}]}}

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

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

ewcommand{
}{\mathrm{N}}

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

Find formulas for $\sum_{d\mid n} \mu(d)\phi(d)$, $\sum_{d \mid n} \mu(d)^2\phi(d)^2$, and $\sum_{d\mid n} \mu(d)/\phi(d)$.

Richard Ganaye · Accepted Answer

\def\imp{\Rightarrow}
\def\gcro{\mbox{[\hspace{-.15em}[}}
\def\dcro{\mbox{]\hspace{-.15em}]}}

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

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

ewcommand{
}{\mathrm{N}}

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

\begin{proof}
As $\mu,\phi$ are multiplicative, so are $\mu \phi, \mu^2\phi^2 ,\mu/\phi$. We deduce from Ex. 2.10 that the three following fonctions $F,G,H$ are multiplicative, defined by
$$F(n) = \sum_{d\mid n} \mu(d)\phi(d), \qquad G(n) = \sum_{d \mid n} \mu(d)^2\phi(d)^2, \qquad H(n) = \sum_{d\mid n} \mu(d)/\phi(d),$$so it is sufficient to compute their values on prime powers $p^k, k\geq 1$.

\begin{align*}
F(p^k) &= \sum_{i=0}^k \mu(p^i)\phi(p^i)\
&= \phi(1) - \phi(p)  = 1 - (p-1) = 2-p
\end{align*}
So $F(n) = \prod_{p\mid n} (2-p)$.

Similarly,
\begin{align*}
G(p^k) &= \sum_{i=0}^k \mu(p^i)^2\phi(p^i)^2\
&= \phi(1)^2 + \phi(p)^2 = 1 + (p-1)^2 = p^2-2p +2
\end{align*}
\begin{align*}
H(p^k) &=  \sum_{i=0}^k \mu(p^i)/\phi(p^i)\
&= 1/\phi(1) -1/\phi(p) = 1 - 1/(p-1) = (p-2)/(p-1)
\end{align*}
\end{proof}

Exercise 2.12

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Exercise 2.12

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