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D) The function given by δ : Z+ −→ R; δ(n) = 1 if n = 1, 0 otherwise. e) The function given by η : Z+ −→ R; η(n) = 1. The set of all real (or complex) arithmetic functions will be denoted by AFR (or AFC ). An arithmetic function ψ is called (strictly) multiplicative if ψ(mn) = ψ(m)ψ(n) whenever gcd(m, n) = 1. 24(b), the Euler function is strictly multiplicative. 1 is strictly multiplicative. An important example is the M¨ obius function µ : Z+ −→ R defined as follows. 19, we have the prime power factorization n = pr11 pr22 · · · prt t , where for each j, pj is a prime, 1 rj and 2 p1 < p2 < · · · < pt .
B) The simplest case is where X ∩ Y = ∅. Then given bijections f : N0 −→ X and g : N0 −→ Y we construct a function h : N0 −→ X ∪ Y by n f 2 h(n) = n−1 g 2 if n is even, if n is odd. Then h is a bijection. If Z = X ∩ Y and Y − Z are both countably infinite, let f : N0 −→ X and g : N0 −→ Y − Z be bijections. Then we define h : N0 −→ X ∪ Y by n f 2 h(n) = n−1 g 2 if n is even, if n is odd. This is again a bijection. The case where one of X − X ∩ Y and Y − X ∩ Y is finite is easy to deal with by the method used for (c).
D) We have θ ∗ ψ(n) = θ(d)ψ(n/d) = d|n ψ(n/d)θ(d) = d|n ψ(k)θ(n/k) = ψ ∗ θ(n). k|n ˜ In each of the groups (AFR , ∗) and (AFC , ∗), the inverse of an arithmetic function θ is θ. Here is an important example. 5. The inverse of η is η˜ = µ, the M¨ obius function. Proof. Recall that η(n) = 1 for all n. 3 we have µ(d)η(n/d) = d|n µ(d) = d|n 1 n = 1, 0 n > 1. 4(c). 6 (M¨obius Inversion). Let f, g : Z+ −→ R (or f, g : Z+ −→ C) be arithmetic functions satisfying f (n) = g(d) (n ∈ Z+ ). d|n Then g(n) = f (d)µ(n/d) d|n (n ∈ Z+ ).
Algebra and number theory, U Glasgow notes by Baker.