Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis

Abstract

For n>1, let G(n)=\sigma(n)/(n log log n), where \sigma(n) is the sum of the divisors of n. We prove that the Riemann Hypothesis is true if and only if 4 is the only composite number N satisfying G(N) \ge \max(G(N/p),G(aN)), for all prime factors p of N and all multiples aN of N. The proof uses Robin's and Gronwall's theorems on G(n). An alternate proof of one step depends on two properties of superabundant numbers proved using Alaoglu and Erd\H{o}s's results.