NTIC Exercises

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Exercises 23.5 Exercises

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1.

Factoring by hand, compute the first 24 values of $λ$ and $ω$ (recall Definition 23.3.4 and Definition 23.3.3).

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2.

Finish the proof that the set of arithmetic functions is a commutative monoid in Theorem 23.4.2.

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3.

Show that if $f = g ⋆ u$ (equivalently, if $g = f ⋆ μ$ ), then $f$ and $g$ are either both multiplicative or both not. Strategy hint: Use Proposition 23.4.11.

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4.

Do enough calculations without using electronic devices to discover a formula (in terms of functions we already know) for the inverse of $N .$

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5.

Do enough calculations without using electronic devices to discover a formula (in terms of functions we already know) for the inverse of $ϕ .$

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6.

Show that the inverse of $λ (n)$ from Definition 23.3.4 is a variant of another of our new functions.

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7.

Can you identify $ω ⋆ μ$ 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.

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8.

Prove Proposition 23.4.10 that using the Dirichlet product on two multiplicative functions stays multiplicative.

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9.

Complete all details of the proof of Theorem 23.4.3 defining inverses under the $⋆$ product.

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10.

Prove Lemma 23.4.12.

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11.

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