On the natural density of integers $n$ for which $σ(kn+r_1) >σ(kn+r_2)$

Xin-qi Luo; Chen-kai Ren

paper

On the natural density of integers $n$ for which $σ (k n + r_{1}) > σ (k n + r_{2})$

arXiv:2311.00295

Abstract

For any positive integer $n$ , let $σ (n) = \sum_{d ∣ n} d$ . In 2020, M. Kobayashi and T. Trudgian showed that the natural density of positive integers n with $σ (k n + r_{1}) \geq σ (k n + r_{2})$ is between 0.053 and 0.055. In this paper, we extend their result. For integers $k > r_{1} > r_{2} \geq 0,$ we provide an estimate on the natural density of positive integers $n$ for which $σ (k n + r_{1}) > σ (k n + r_{2})$ . We also calculate some special cases with certain $k, r_{1}$ and $r_{2}$ . We also compute explicit bounds for specific $k, r_{1}, r_{2}$ to illustrate the variation of density.

9 pages

On the natural density of integers n for which σ(kn+r1​)>σ(kn+r2​)

Abstract

On the natural density of integers $n$ for which $σ (k n + r_{1}) > σ (k n + r_{2})$