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Number Theory

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Moderate

Mobius Function and Divisor Function

NUMTH-TYEZY$

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

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

Which of the following formulas are correct for all natural number $n$?

Select ALL that apply.

A

$\sum_{d|n} \mu(d) = 1$.

B

$\sum_{d|n} \tau(d) = \tau(n)$.

C

$\sum_{d|n} \mu(d)\tau\left(\frac nd\right) = 1$.

D

$\sum_{d|n} \mu(d) \tau(d) = \begin{cases} 1 &\mbox{ if }n=1 \\\ 0 &\mbox{ otherwise}\end{cases} $