1.5. Sums over divisors 17 operation on arithmetic functions is the Dirichlet convolution. If f and g are arithmetic functions then (f ∗ g)(n) = d|n f(d)g n d = km=n f(k)g(m) is their Dirichlet convolution. It is a straightforward exercise to see that under addition and Dirichlet convolution, the arithmetic functions form a commutative ring with multiplicative neutral element e where e(1) = 1 and e(n) = 0 for n ≥ 2. This is called the Dirichlet ring. Denoting the constant function equal to 1 by 1 we note d = 1 ∗ 1 as an example of Dirichlet convolution. Another convolution identity is 1 ∗ Λ = log. This is easily proved by observing that (1 ∗ Λ)(n) = d|n Λ(d) = pk|n log(p) = log(n), since Λ is zero off the prime powers. This can replace the Legendre identity as the point of entry for the method of Chebyshev. Proposition 1.12. If f and g are multiplicative, so is f ∗ g. Proof. If gcd(m, n) = 1, then the divisors d|mn are precisely those positive integers of the form d = bc where b|m and c|n. Hence (f ∗ g)(mn) = d|mn f(d)g mn d = b|m,c|n f(bc)g mn bc = b|m,c|n f(b)f(c)g m b g n c = b|m f(b)g m b c|n f(c)g n c = (f ∗ g)(m)(f ∗ g)(n) by the multiplicativity of f and g. Since 1 is multiplicative, Proposition 1.12 shows that d is also multi- plicative. The sum-of-divisors function σ(n) = d|n d is given by the Dirichlet convolution σ = 1∗id, so σ is multiplicative, because id is. Part of the significance of Proposition 1.12 is that for Dirichlet con- volutions of multiplicative functions it affords a straightforward means of calculation it is enough to calculate their values on prime powers. Let

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