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Arithmetical Functions of a Greatest Common Divisor, III. Cesàro's Divisor Problem

Published online by Cambridge University Press:  18 May 2009

Eckford Cohen
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Let σ1(n) denote the sum of the tth powers of the divisors of n, σ(n) = σ1(n). Also place

where 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).

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 5 , Issue 2 , July 1961 , pp. 67 - 75
Copyright
Copyright © Glasgow Mathematical Journal Trust 1961

References

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