Section23.5Exercises

  1. Finish the proof that the set of arithmetic functions is a commutative monoid.
  2. Prove that using the Dirichlet product on two multiplicative functions stays multiplicative.
  3. Show that if \(f=g\star u\) (equivalently, if \(g=f\star \mu\)), then \(f\) and \(g\) are either both multiplicative or both not. Strategy hint: Use the fact about multiplicativity of inverses in the set of arithmetic functions under \(\star\).
  4. Show that the inverse of the function \(\lambda(n)\) above is a variant of another of our new functions.
  5. Can you identify \(\omega\star\mu\) as anything familiar?
  6. Come up with another good exercise for this chapter.
  7. Come up with two additional good exercises for this chapter, different from the one in the previous exercise.