1.6. The hyperbola method 23 Another approach to study the growth of the rapidly fluctuating arith- metic function d(n) is to consider a local average such as 1 h x−hn≤x d(n). The fluctuations of d(n) are smoothed out by the process of averaging. Then 1 h x−hn≤x d(n) = D(x) D(x h) h where D(x) is the summatory function of the divisor function. The use of local averaging to study the growth of arithmetic functions can be traced back to article 301 in the Disquisitiones Arithmeticae. Gauss was interested in the growth of the class number and the number of genera for binary quadratic forms, as (irregularly fluctuating) arithmetic functions of the dis- criminant. He quoted results on the rate of growth of their local averages, but judged the proofs to be too difficult to include in the Disquisitiones. To calculate a local average by differencing an estimate for the associated summatory function is not always efficient. When h is small, one is apt to run into the same kind of problem as one does in numerics when subtracting floating-point numbers that are nearly equal. The bijection d n/d on the set of divisors d of an integer n is called the Dirichlet interchange. Since d n is equivalent to n/d n it is clear that d(n) = 2 nd|n 1 unless n is a square. In the latter case the divisor n is missing and it is necessary to add 1 on the right-hand side. Applying this formula to the definition of D(x) gives D(x) = [ x] + n≤x 2 nd|n 1 = [ x] + 2 d≤ x d2kd≤x 1 = [ x] + 2 d≤ x x d d = 2 m≤ x x m [ x]2. The latter formula is due to D. F. E. Meissel. Proposition 1.16. The estimate D(x) = x log(x) + (2γ 1)x + O(x1/2) holds.
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