20 1. Arithmetic Functions 1 for n = k and is zero otherwise. The calculation n≤x (e1(n) − 2e2(n))T x n = n≤x (e1(n) − 2e2(n)) m≤x/n ψ x/n m = k≤x ψ x k d|k (e1(d) − 2e2(d)) = k≤x (−1)k−1ψ x k may be taken as the framework of the proof of Proposition 1.1. It can be generalized by replacing e1−2e2 with a more complicated linear combination f of e1, e2,..., eN. Then 1 ∗ f is required to take only the values 0, ±1, and the nonzero values of this function should start with (1 ∗ f)(1) = 1 and alternate. Chebyshev chose the linear combination f = e1 −e2 −e3 −e5 +e30 to obtain his better estimates. We exhibit an example of the use of M¨ obius inversion to establish an arithmetically significant estimate. We find a quite precise bound for the error term in an asymptotic estimate for the summatory function Φ(x) def = n≤x φ(n) of the Euler totient. The convolution identity of Gauss yields n≤x Φ x n = n≤x d≤x/n φ(d) = nd≤x φ(d) = m≤x d|m φ(d) = m≤x (1 ∗ φ)(m) = m≤x m = 1 2 [x]([x] + 1), and then Φ(x) = n≤x μ(n) 1 2 x n x n + 1 = 1 2 n≤x μ(n) x n + O(1) x n + O(1) = x2 2 n≤x μ(n) n2 + O⎝x ⎛ n≤x 1 n ⎞ ⎠ = x2 2 ∞ n=1 μ(n) n2 + O x2 nx 1 n2 + O(x log(x)) = x2 2 ∞ n=1 μ(n) n2 + O(x log(x)) by the second M¨ obius inversion formula. Absolutely convergent series may be multiplied together to yield absolutely convergent series, and so ∞ m=1 1 m2 ∞ n=1 μ(n) n2 = ∞ N=1 1 N 2 mn=N 1·μ(n) = ∞ N=1 e(N) N 2 = 1.
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