Geodesic
Arithmetical Functions of a Greatest Common Divisor, III. Cesàro's Divisor Problem
Glasgow mathematical journal, Tome 5 (1961) no. 2, pp. 67-75

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Let σ1(n) denote the sum of the tth powers of the divisors of n, σ(n) = σ1(n). Also placewhere γ is Euler's constant, ζ(s) is the Riemann ζ-function and x ≧ 2. The function Δ(x) is the remainder term arising in the divisor problem for σ((m, n)). Cesàro proved originally [1], [6, p. 328] that Δ(x) = o(x2 log x). More recently in I [2, (3.14)] it was shown by elementary methods that . This estimate was later improved to in II [3, (3.7)]. In the present paper (§ 3) we obtain a much more substantial reduction in the order of Δ(x), by showing that Δ(x) can be expressed in terms of the remainder term in the classical Dirichlet divisor problem. On the basis of well known results for this problem, it follows easily that . The precise statement of the result for σ((m, n)) is contained in (3.2). 
Arithmetical Functions of a Greatest Common Divisor, III. Cesàro's Divisor Problem. Glasgow mathematical journal, Tome 5 (1961) no. 2, pp. 67-75. doi: 10.1017/S2040618500034328
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