Say $f$ is multiplicative and $m = p^{\alpha}q^{\beta}$ where $p,q$ are prime. Then do we have $f(m) = f(p^{\alpha})f(q^{\beta})$? If so why?. I know this hold when the powers are 1 but have not been given an explanation for higher powers.

  • 3
    $\begingroup$ The standard definition of a multiplicative function says that $\gcd(a,b)=1\implies f(ab)=f(a)f(b)$. So, in your case, all you need is for the primes $p.q$ to be distinct.; $\endgroup$ – lulu May 10 at 10:38

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.