24 1. Arithmetic Functions Proof. We have D(x) = 2 n≤ x x n [ x]2 = 2 n≤ x x n + O(1) ( x + O(1))2 = 2x(log( x) + γ + O(1/ x)) x + O(x1/2) = x log(x) + (2γ 1)x + O(x1/2) by Proposition 1.7. This estimate is due to J. P. G. Lejeune Dirichlet. It implies that the arithmetic average of d(n) over the range 1 n x is asymptotic to log(x) as x +∞. One says that d(n) has average order log(x). From Proposition 1.15 it is easy to see that d(n) is sometimes larger than any fixed power of log(n). But Proposition 1.16 implies that d(n) is only rarely so large. The notation Δ(x) = D(x) x log(x) (2γ 1)x is traditional for the error term in the estimate in Proposition 1.16. The problem of bounding Δ(x) is known as the Dirichlet divisor problem. More precisely, the divisor problem is to find the least ϑ for which an estimate Δ(x) = O(xϑ+ε) holds for all ε 0. The result just proved shows that ϑ 1/2. For x large and h rather smaller than x, say h x/2, one obtains 1 h x−hn≤x d(n) = D(x) D(x h) h = x log(x) + (2γ 1)x + Δ(x) h (x h) log(x h) + (2γ 1)(x h) + Δ(x h) h = log(x) + + O h x + O x1/2 h by the estimate Δ(x) = O(x1/2). The error is a sum of two terms, one of which dominates when h is large and the other when h is small. In such situations one would usually try to choose the parameter optimally to obtain a small error term overall. Minimizing h/x + x1/2/h over h for x fixed, one sees that h = x3/4 is an optimal choice. Thus 1 h x−hn≤x d(n) = log(x) + + O(x−1/4), h = x3/4. The asymptotic law of growth log(x) + for the local average of d(n) was Dirichlet’s main application in his 1849 paper on the divisor problem. The
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