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 Erdős's results.
 Publication:

arXiv eprints
 Pub Date:
 October 2011
 arXiv:
 arXiv:1110.5078
 Bibcode:
 2011arXiv1110.5078C
 Keywords:

 Mathematics  Number Theory;
 Mathematics  History and Overview;
 11M26 (Primary) 11A41;
 11Y55 (Secondary)
 EPrint:
 11 pages, 1 table, clarified Proposition 4, added reference 4