Exercise 2.21

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}}

Define $\land(n) = \log p$ if $n$ is a power of $p$ and zero otherwise. Prove that $\sum_{d \mid n} \mu(n/d)\log d = \land (n)$. [Hint: First calculate $\sum_{d \mid n} \land(d)$ and then apply the M"{o}bius inversion formula.]

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}
$$
\left\{
\begin{array}{cccl}
  \land(n)& =  & \log p & \mathrm{if}\  n =p^\alpha,\ \alpha \in \N^*  \
  &  = &   0 & \mathrm{otherwise }.
\end{array}
ight.
$$
Let $n = p_1^{\alpha_1}\cdots p_t^{\alpha_t}$ be the decomposition of $n$ in prime factors. As $\land(d) = 0$ for all factors of $n$, except for $d = p_j^i, i>0, j=1,\ldots t$,
\begin{align*}
\sum_{d \mid n} \land(d)&= \sum_{i=1}^{\alpha_1} \land(p_1^{i}) + \cdots+ \sum_{i=1}^{\alpha_t} \land(p_t^{i})\ 
&= \alpha_1 \log p_1+\cdots + \alpha_t \log p_t\
&= \log n
\end{align*}
By M"{o}bius Inversion Theorem,
$$\land(n) = \sum_{d \mid n} \mu\left (\frac{n}{d}ight ) \log d.$$
\end{proof}

Exercise 2.21

Answers

Comments

Exercise 2.21

Answers

Comments

Add answer