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If an integer $n>1$ has prime factorization $n=p_1^{k_1}p_2^{k_2}p_3^{k_3}\cdots p_r^{k_r}$ which of the following are true?

  1. $\sum_{d|n}\mu(d)\tau(d)=1$
  2. $\sum_{d|n}\mu(d)\tau(d)=p_1p_2\cdots p_r$
  3. $\sum_{d|n}\frac{\mu(d)}{d}=\sum_{i=1}^{r}\left( 1-\frac{1}{p_i}\right)$
  4. $\sum_{d|n}\mu(d)d=\prod_{i=1}^{r}(1-p_i)$

using $\phi(n)=n\sum_{d|n}\frac{\mu(d)}{d}$ I can show that 3 is not true. But for the rest I don't have any idea. I think we have to use Mobius Inversion formula, but don't know how? Any help please. Thanks.

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    Well, 1. and 2. can't both be true, can they? So why not plug in your favorite value of $n$, you'll eliminate at least one of the alternatives. In other words, don't just sit there – try something! – 2017-02-16
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    Yes @GerryMyerson I am trying. Actually I was thinking of proving them, but completely forgot about trial and error method. Let me check. – 2017-02-16

1 Answers 1

1

For a full explanation:

1 and 2: The only nonzero terms of $\sum_{d|n}\mu(d)\tau(d)$ come from $d$ being a product of distinct primes. There are $\binom rs$ divisors which are a product of $s$ distinct primes, and each of these contributes $(-1)^s2^s$. So $$\sum_{d|n}\mu(d)\tau(d)=\sum_{s=0}^r\binom rs(-2)^s=(1-2)^r=(-1)^r.$$

3: as you say, $\phi(n)=n\sum_{d|n}\frac{\mu(n)}d$ gives the RHS as the product of those terms, not the sum, so this is wrong.

4: Since $d\mu(d)$ is multiplicative, so is $\sum_{d|n}d\mu(d)$, and the RHS is also multiplicative, so it enough to check it is correct when $n=p^k$. In this case only the terms corresponding to $d=1$ and $d=p$ are nonzero, and this gives $f(p^k)=1-p$, as required.