Simplifying expression Mobius Function

Question

Can anybody help me simplify this expression using Mobius Inversion Formula or any other result in order to calculate F(3500) in a simple way?? $$F(n)=\sum_{d\mid n} \mu(d)d$$

Answer 1

It is the product $$f(n)=(1-p_1)(1-p_2)\cdots(1-p_k)$$ where the $p_i$ are the distinct prime divisors of $d.$

This is because $g(n)=\mu(n)n$ is multiplicative, and hence so is $F,$ and $F(p^k)=1-p$ for prime $p.$

In your case, $3500$ has prime divisors $2,5,$ and $7$. So:

$$F(3500)=(1-2)(1-5)(1-7)=-24$$

You could write it as $F(n)=n\sum_{d\mid n} \mu(d)\frac{1}{n/d}.$ Then you get, by Möbius inversion: $$\frac{1}{n}=\sum_{d\mid n}\frac{F(d)}{d}$$

Or:

$$1=\sum_{d\mid n}\frac{n}{d}F(d)$$

That won't help you compute $F$ much however.