Robin's Theorem, Primes, and a new elementary reformulation of the Riemann Hypothesis

Geoffrey Caveney (1) , Jean-Louis Nicolas (1) , Jonathan Sondow (1)

1 ICJ - Institut Camille Jordan

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) >= max( G(N/p),G(aN)), for all prime factors factors p of N and each positive integer a. 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 Erdös's results.

Domaines: Théorie des nombres [math.NT]

Geoffrey Caveney, Jean-Louis Nicolas, Jonathan Sondow. Robin's Theorem, Primes, and a new elementary reformulation of the Riemann Hypothesis. Integers : Electronic Journal of Combinatorial Number Theory, 2011, pp.11. ⟨hal-00745160⟩