26 1. Arithmetic Functions x x uv x u v Figure 2. Lattice points in the divisor problem Proof. We have n≤x (f ∗ g)(n) = n≤x km=n f(k)g(m) = k≤y f(k) km≤x g(m) + ky km≤x f(k)g(m) = k≤y f(k)G x k + m≤x/y g(m) yk≤x/m f(k) = k≤y f(k)G x k + m≤x/y g(m)⎝ ⎛ k≤x/m f(k) − k≤y f(k)⎠ ⎞ = k≤y f(k)G x k + m≤x/y g(m)F x m − F (y)G x y for any y with 1 ≤ y ≤ x. Questions about divisors more delicate than their gross count have also been studied. The monograph Divisors by R. R. Hall and G. Tenenbaum is a good source for information of this kind.
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