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

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11 Pages
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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.

  • used robin's

  • only extraor- dinary

  • swedish mathematician

  • srinivasa ramanujan proved

  • numbers

  • euler-mascheroni constant

  • robin's inequality

  • yields lim

  • all known numbers

  • function ?


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ROBIN’S THEOREM, PRIMES,
REFORMULATION OF THE

AND A NEW ELEMENTARY
RIEMANN HYPOTHESIS

Geoffrey Caveney
7455 North Greenview #426, Chicago, IL 60626, USA
rokirovka@gmail.com

Jean-Louis Nicolas
Universit´ de Lyon; CNRS; Universit´ Lyon 1;
Institut Camille Jordan, Math´matiques,
21 Avenue Claude Bernard, F-69622 Villeurbanne cedex, France
nicolas@math.univ-lyon1.fr

Jonathan Sondow
209 West 97th Street #6F, New York, NY 10025, USA
jsondow@alumni.princeton.edu

Abstract
Let
σ(n)
G(n() =n >1),
nlog logn
whereσ(n) is the sum of the divisors ofn. Weprove that the Riemann Hypothesis
is true if and only if 4 is the only composite numberNsatisfying
G(N)≥max (G(N/p), G(aN)),
for all prime factorspofNand each positive integeraproof uses Robin’s. The
and Gronwall’s theorems onG(n). Analternate proof of one step depends on two
propertiesofsuperabundantnumbersprovedusingAlaogluandErd˝os’sresults.

1. Introduction

Thesum-of-divisors functionσis defined by
X
σ(n) :=d.
d|n
For example,σ(4) = 7 andσ(pn) = (p+ 1)σ(n), ifpis a prime not dividingn.
In 1913, the Swedish mathematician Thomas Gronwall [4] found the maximal
order ofσ.