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).

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