1.6. The hyperbola method 25 asymptotic estimate for the local average may be improved in two different ways: By shortening the interval or reducing the error term. Modifying the estimate as it stands by shortening the interval as much as possible, we lose the error term and obtain the asymptotic estimate 1 h x−hn≤x d(n) ∼ log(x) + 2γ, h = o(x1/2/ log(x)). Using a better estimate in the divisor problem, the estimate for the local average may be improved both by reducing the error term and shortening the interval. One may also prove Proposition 1.16 from Dirichlet’s formula D(x) = dk≤x 1 = n≤x x n . That D(x) = x log(x) + O(x) is immediate from this formula. The better estimate for the error term may be obtained by observing that the sum equals the number of integer lattice points in the region of the uv-plane given by the inequalities u ≥ 1, v ≥ 1 and uv ≤ x. One can then recover the formula of Meissel by observing that the union of the two subregions obtained by imposing the inequalities u ≤ √ x and v ≤ √ x equals the original region, while their intersection equals the square given by 1 ≤ u ≤ √ x and 1 ≤ v ≤ √ x. The interpretation of D(x) in terms of the number of lattice points under a hyperbola is very important for more advanced work on the divisor problem. See Figure 2 on page 26 for an illustration. The Dirichlet interchange and the approach to the Meissel formula based on counting lattice points under a hyperbola are closely connected. The technique is usually called the Dirichlet hyperbola method. It has other applications and so we exhibit a more general formulation due to H. G. Diamond. Proposition 1.17 (Dirichlet hyperbola method). If f is an arithmetic func- tion with summatory function F and g an arithmetic function with summa- tory function G then n≤x (f ∗ g)(n) = k≤y f(k)G x k + m≤x/y g(m)F x m − F (y)G x y for 1 ≤ y ≤ x.

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