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Publié par | profil-urra-2012 |
Nombre de lectures | 28 |
Langue | English |
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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.