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Section23.5Exercises

1

Factoring by hand, compute the first 24 values of \lambda and \omega.

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 23.4.9.

4

Do enough calculations the old-fashioned way to discover a name for the inverse of N.

5

Do enough calculations the old-fashioned way to discover a name for the inverse of \phi.

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 23.4.8 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.

11

Come up with another good exercise for this chapter and have a friend try it!