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
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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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