1. PANORAMA OF ARITHMETIC FUNCTIONS 5 Now we are going to give some basic examples of arithmetic functions. We begin by exploiting Dirichlet convolutions. The first one is the divisor function τ = 1 1, τ(n) = d|n 1, ζ2(s) = 1 τ(n)n−s. This is multiplicative with τ(pα) = α + 1. We have ζ4(s) ζ(2s) = 1 τ(n)2n−s ζ3(s) ζ(2s) = 1 τ(n2)n−s ζ3(s) = 1 d2m=n τ(m2) n−s. Note that (1.13) d2m=n τ(m2) = klm=n 1 = τ3(n), say. Next we get ζ2(s) ζ(2s) = 1 2ω(n)n−s where ω(n) denotes the number of distinct prime divisors of n, so 2ω(n) is the number of squarefree divisors of n. The characteristic function of squarefree numbers is μ(n) = μ2(n) = d2|n μ(d), its Dirichlet series is ζ(s) ζ(2s) = 1 |μ(n)|n−s = p 1 + 1 ps . Inverting this we get the generating series for the Liouville function λ(n) ζ(2s) ζ(s) = 1 λ(n)n−s. Note that λ(n) = (−1)Ω(n) where Ω(n) is the total number of prime divisors of n (counted with the multiplic- ity). The Euler ϕ-function is defined by ϕ(n) = |(Z/nZ)∗| it is the number of reduced residue classes modulo n. This function satisfies ϕ(n) = n p 1 1 p = n d|n μ(d) d . Hence ζ(s 1) ζ(s) = 1 ϕ(n)n−s.
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