Suppose that fngn is a M obius pair of arithmetic functions

Suppose that {f(n),g(n)} is a M ?obius pair of arithmetic functions. By a theorem we proved in class we know that if g is multiplicative, then f must be multiplicative too. Suppose that g is totally multiplicative; must f be totally multiplicative as well? Prove or give a counter-example.

Solution

A totally multiplicative function g(n) is defined completely by its values on primes.

Let (n) be defined by setting f(2^k) =2^k and f(p)=1 for all odd primes.

Let f(n) be such that {g,f} form a Mobius pair.

Then f(2) =1+2=3 and f(4) =1+2+4=7.

So f is not totally multiplicative.