Exercises 23.5 Exercises
1.
Factoring by hand, compute the first 24 values of \(\lambda\) and \(\omega\) (recall Definition 23.3.4 and Definition 23.3.3).
2.
Finish the proof that the set of arithmetic functions is a commutative monoid in Theorem 23.4.2.
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 Proposition 23.4.11.
4.
Do enough calculations without using electronic devices to discover a formula (in terms of functions we already know) for the inverse of \(N\text{.}\)
5.
Do enough calculations without using electronic devices to discover a formula (in terms of functions we already know) for the inverse of \(\phi\text{.}\)
6.
Show that the inverse of \(\lambda(n)\) from Definition 23.3.4 is a variant of another of our new functions.
7.
Can you identify \(\omega\star\mu\) as anything familiar? (Recall Definition 23.3.3.) If yes, then try to prove it; if not, explain why you think it is new to us.
8.
Prove Proposition 23.4.10 that using the Dirichlet product on two multiplicative functions stays multiplicative.
9.
Complete all details of the proof of Theorem 23.4.3 defining inverses under the \(\star\) product.
10.
Prove Lemma 23.4.12.
11.
Come up with another good exercise for this chapter and have a friend try it!