Robin’s Criterion on Superabundant Numbers

The Riemann hypothesis is the assertion that all non-trivial zeros are complex numbers with real part 1/2. It is considered by many to be the most important unsolved problem in pure mathematics. There are several statements equivalent to the famous Riemann hypothesis. Robin's criterion states that the Riemann hypothesis is true if and only if the inequality σ(n) < eγ · n · log log n holds for all natural numbers $n > 5040$, where σ(n) is the sum-of-divisors function of n, γ ≈ 0.57721 is the Euler-Mascheroni constant and log is the natural logarithm. We require the properties of superabundant numbers, that is to say left to right maxima of n ↦ σ(n)/n. In this note, using Robin's criterion on superabundant numbers, we prove that the Riemann hypothesis is true.