18 1. Arithmetic Functions us, for example, calculate the Dirichlet convolution 1 ∗ φ. Both factors are multiplicative, so the calculation (1 ∗ φ)(pα) = pβ|pα 1 · φ(pβ) = 1 + α β=1 (p − 1)pβ−1 = pα yields the convolution identity 1 ∗ φ = id of Gauss. The M¨ obius mu-function μ is the unique multiplicative arithmetic func- tion with values μ(p) = −1 on the primes p, and values μ(pk) = 0 on the prime powers pk with k ≥ 2. The M¨ obius function has a strong combinato- rial flavor. It is closely connected to the principle of inclusion and exclusion, and to the fact that the integers form a partially ordered set under the re- lation of divisibility. The importance of the M¨ obius function is due to the convolution identity d|n μ(d) = e(n). There are many ways to establish that 1 ∗ μ = e, but the quickest is to recall that μ is multiplicative. Then so is 1 ∗ μ, and thus (1 ∗ μ)(pα) = 1 + (−1) + 0 + 0 + · · · = 0 yields (1 ∗ μ)(n) = 0 for all n ≥ 2. Proposition 1.13 (First M¨ obius inversion formula). If g = 1 ∗ f then f = μ ∗ g and conversely. Proof. If g = 1 ∗ f, then μ ∗ g = μ ∗ (1 ∗ f) = (μ ∗ 1) ∗ f = e ∗ f = f, and if f = μ ∗ g then 1 ∗ f = 1 ∗ (μ ∗ g) = (1 ∗ μ) ∗ g = e ∗ g = g. This shows that 1 is a unit in the Dirichlet ring, and μ is its multiplicative inverse. An arithmetic function f is a unit if and only if f(1) = 0. Under this condition a Dirichlet inverse g for f may be constructed incrementally from g(n) = e(n) − n=d|n f n d g(d). The relation f(1)g(1) = e(1) = 1 shows that the condition f(1) = 0 is necessary. Constructing an explicit Dirichlet inverse is usually infeasible, except in the very important case when f is multiplicative. The first M¨obius inversion formula is often formulated as the statement “f(n) = d|n μ(d)g n d if and only if g(n) = d|n f(d)” about divisor sums.

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