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VARIATIONS ON A THEOREM OF DAVENPORT CONCERNING ABUNDANT NUMBERS

2013
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Bulletin of the Australian Mathematical Society
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Let σ(n) = ∑_d | nd be the usual sum-of-divisors function. In 1933, Davenport showed that that n/σ(n) possesses a continuous distribution function. In other words, the limit D(u):= _x→∞1/x∑_n ≤ x, n/σ(n) ≤ u 1 exists for all u ∈ [0,1] and varies continuously with u. We study the behavior of the sums ∑_n ≤ x, n/σ(n) ≤ u f(n) for certain complex-valued multiplicative functions f. Our results cover many of the more frequently encountered functions, including φ(n), τ(n), and μ(n). They also apply

doi:10.1017/s0004972713000695
fatcat:7ahrup3ph5ff7gtzly5suun3nm