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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.
- Divisor function
- Primary: 11B05
- Secondary: 11A25
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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.