Robin also proved, unconditionally, that the inequality:   σ ( n ) < e γ n log ⁡ log ⁡ n + 0.6483   n log ⁡ log ⁡ n {\displaystyle \ \sigma (n)<e^{\gamma }n\log \log n+{\frac {0.6483\ n}{\log \log n}}} holds for all n ≥ 3.
Robin bevisade också att olikheten   σ ( n ) < e γ n log ⁡ log ⁡ n + 0.6483   n log ⁡ log ⁡ n {\displaystyle \ \sigma (n)<e^{\gamma }n\log \log n+{\frac {0.6483\ n}{\log \log n}}} gäller för alla n ≥ 3.