6 1. PANORAMA OF ARITHMETIC FUNCTIONS A different class of arithmetic functions (very important for the study of prime numbers) is obtained by differentiating relevant Dirichlet series −Df(s) = ∞ 1 f(n)(log n)n−s. In particular −ζ (s) = ∞ 1 (log n)n−s. Since L(n) = log n is additive, we have the formula L · (f ∗ g) = (L · f) ∗ g + f ∗ (L · g) which says that the multiplication by L is a derivation in the Dirichlet ring of arithmetic functions. By the Euler product we have (1.14) log ζ(s) = ∞ l=1 p l−1p−ls. Hence differentiating we get (1.15) − ζ ζ (s) = ∞ 1 Λ(n)n−s with Λ(n) (popularly named von Mangoldt function) given by (1.16) Λ(n) = log p, if n = pα, α 1 0, otherwise. From the left side of (1.15) we get (1.17) Λ = μ ∗ L, Λ(n) = d|n μ(d) log n d = − d|n μ(d) log d. Hence, by M¨ obius inversion we get (1.18) L = 1 ∗ Λ, log n = d|n Λ(d). Similarly we define the von Mangoldt functions Λk of any degree k 0 by (1.19) Λk = μ ∗ Lk, Λk(n) = d|n μ(d) log n d k . We have (1.20) Lk = 1 ∗ Λk, (log n)k = d|n Λk(d). Note that Λ0 = δ, Λ1 = Λ, and we have the recurrence formula (1.21) Λk+1 = L · Λk + Λ ∗ Λk. This follows by writing Λk+1(n) = d|n μ(d) log n d k (log n − log d) and using (1.18).
Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2014 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.
