On the density of ranges of generalized divisor functions

Colin Defant
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 21, 2015, Number 3, Pages 80–87
Full paper (PDF, 173 Kb)

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Authors and affiliations

Colin Defant
Department of Mathematics, University of Florida
1400 Stadium Rd., Gainesville, FL 32611
United States

Abstract

The range of the divisor function σ−1 is dense in the interval . However, although the range of the function σ−2 is a subset of the interval , we will see that the range of σ−2 is not dense in . We begin by generalizing the divisor functions to a class of functions σt for all real t. We then define a constant η ≈ 1.8877909 and show that if r ∈ (1, ∞), then the range of the function σ−r is dense in the interval if and only if r ≤ η. We end with an open problem.

Keywords

  • Density
  • Divisor function

AMS Classification

  • Primary: 11B05
  • Secondary: 11A25

References

  1. Dusart, P. (2010) Estimates of some functions over primes without R.H., arXiv:1002.0442.
  2. Laatsch, R. (1986) Measuring the abundancy of integers. Math. Mag., 59(2), 84–92.
  3. Wolfram Research, Inc., Mathematica, Version 9.0, Champaign, IL (2012)

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Cite this paper

Defant, C. (2015). On the density of ranges of generalized divisor functions. Notes on Number Theory and Discrete Mathematics, 21(3), 80-87.

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