If Robin's inequality, i.e., σ(n) < e^γ n log log n, is satisfied for all integers n > 5040, then the Riemann Hypothesis is true. Robin's inequality has been proven for various infinite families of integers greater than 5040: 5-free integers, 7-free integers, and recently improved to 11-free integers. We use new bounds to demonstrate that the inequality in fact holds for 15-free integers, and discuss obstacles to further improvement.