Exercise 2.20

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}}

Prove that $\prod_{d \mid n} d = n^{
u(n)/2}$.

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}
Let $$n = p_1^{\alpha_1}\cdots p_k^{\alpha_k}$$ be the decomposition of $n$ in prime factors.
\begin{align*}
\left( \prod_{d \mid n}d ight)^2 &=  \prod_{d \mid n} d \  \prod_{d \mid n} d \
&= \prod_{d \mid n} d  \  \prod_{\delta \mid n} \frac{n}{\delta} \qquad (\delta = n/d)\
&=n^{
u(n)}  \prod_{d \mid n} d  \  \prod_{d \mid n} \frac{1}{d} \
&=n^{
u(n)}
\end{align*}
Conclusion:
$$\prod_{d \mid n} d = n^{\frac{
u(n)}{2}}.$$
\end{proof}

Exercise 2.20

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Exercise 2.20

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