Colin Defant

Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132

Volume 21, 2015, Number 3, Pages 80—87

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## Details

### 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 σ_{t} 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

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

## Related papers

## Cite this paper

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

ChicagoDefant, Colin. “On the Density of Ranges of Generalized Divisor Functions.” Notes on Number Theory and Discrete Mathematics 21, no. 3 (2015): 80-87.

MLADefant, Colin. “On the Density of Ranges of Generalized Divisor Functions.” Notes on Number Theory and Discrete Mathematics 21.3 (2015): 80-87. Print.