ROBIN'S THEOREM PRIMES AND A NEW ELEMENTARY REFORMULATION OF THE RIEMANN HYPOTHESIS

ROBIN'S THEOREM, PRIMES, AND A NEW ELEMENTARY REFORMULATION OF THE RIEMANN HYPOTHESIS Geoffrey Caveney 7455 North Greenview _426, Chicago, IL 60626, USA Jean-Louis Nicolas Universite de Lyon; CNRS; Universite Lyon 1; Institut Camille Jordan, Mathematiques, 21 Avenue Claude Bernard, F-69622 Villeurbanne cedex, France Jonathan Sondow 209 West 97th Street _6F, New York, NY 10025, USA Abstract Let G(n) = ?(n) n log log n (n > 1), where ?(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 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 Erdo˝s's results. 1. Introduction The sum-of-divisors function ? is defined by ?(n) := ∑ d|n d.

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