Variations on a theorem of Davenport concerning abundant numbers

Authors

  • E. Jennings University of Georgia
  • P. Pollack University of Georgia
  • L. Thompson University of Georgia

Keywords:

abundant number, distribution function, mean values of multiplicative functions, sum-of-divisors function

Abstract

DOI: Let \(\sigma(n)\) denote the sum of the positive divisors of \(N\). In 1933, Davenport showed that \(n/\sigma(n)\) possesses a continuous distribution function. We study the behavior of analogous weighted distributions involving certain complex-valued multiplicative functions. Our results cover many of the more frequently encountered functions, including \(\sigma(n)\), \(\tau(n)\) and \(\mu(n)\). They also apply to the representation function for sums of two squares, leading again to a continuous distribution function. 10.1017/S0004972713000695

Published

2014-03-25

Issue

Vol. 89 No. 3 (2014)

Section

Articles
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