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Number Theory

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An Identity Between Arithmetic Functions

NUMTH-BBEG2Y

Let $\phi(n)$ be the Euler-Phi function. Define $\tau(n):=\sum_{d|n} 1$ by the number of divisors of a natural number $n$ and $\mu(n)$ by the Mobius function, which is given by $\mu(1)=1$ and:

$$ \mu(n):=\begin{cases} (-1)^r &\mbox{ if }n=p_1p_2 \cdots p_r \mbox{ for distinct primes }p_1, \ldots, p_r \\\ 0 &\mbox{ otherwise.}\end{cases} $$

Dirichlet convolution of two arithmetic functions $f$ and $g$ is defined by:

$$ f \ast g := \sum_{d|n} f(d) g\left(\frac nd\right) $$

Also, define $\bf 1 \rm (n) := 1$ for all natural numbers $n$, and:

$$\delta_1(n) := \begin{cases} 1 &\mbox{ if } n=1 \\\ 0&\mbox{ otherwise} \end{cases}$$

Complete the following identity:
$$ \sum_{d|n} \frac d{\phi(d)} \mu\left(\frac nd\right) = $$

A

$\frac{\mu(n)}{\phi(n)}$

B

$\frac 1{\phi(n)}$

C

$\frac{\mu(n)}{\phi(n)^2}$

D

$\frac{\mu(n)^2 }{\phi(n)}$

E

$\frac{\mu(n)^2}{\phi(n)^2}$