```Date: Jun 19, 2013 8:56 AM
Author: David Bernier
Subject: Re: abundant numbers, Lagarias criterion for the Riemann Hypothesis

On 06/13/2013 12:52 PM, David Bernier wrote:> On 06/13/2013 10:38 AM, David Bernier wrote:>> I've been looking for abundant numbers, a number `n' whose>> sum of divisors sigma(n):= sum_{d dividing n} d>> is large compared to `n'.>>>> One limiting bound, assuming the Riemann Hypothesis,>> is given by a result of Lagarias:>>>> whenener n>1,  sigma(n) < H_n + log(H_n)*exp(H_n) ,>> where H_n := sum_{k=1 ... n} 1/k .>>>> Cf.:>> < http://en.wikipedia.org/wiki/Harmonic_number#Applications > .>>>> The measure of "abundance" I use, for an integer n>1, is>> therefore:>>>> Q =  sigma(n)/[ H_n + log(H_n)*exp(H_n) ].>>>> For n which are multiples of 30, so far I have the>> following `n' for which the quotient of "abundance">> Q [a function of n] surpasses  0.958 :>>>> n             Q>> ----------------------->> 60        0.982590>> 120       0.983438>> 180       0.958915>> 360       0.971107>> 840       0.964682>> 2520      0.978313>> 5040      0.975180>> 10080     0.959301>> 55440     0.962468>> 367567200 0.958875>>>> What is known about lower bounds for>>>> limsup_{n-> oo}  sigma(n)/[ H_n + log(H_n)*exp(H_n) ]  ?>> I know there's Guy Robin earlier and, I believe, Ramanujan> who worked on "very abundant" numbers ...>> n = 2021649740510400   with Q = 0.97074586,>> almost as "abundantly abundant" as n=360, with Q = 0.971107>> sigma(2,021,649,740,510,400) = 12,508,191,424,512,000I've used PARI/gp to find whole numbers with as largea "quotient of abundance" Q as possible, and it hastaken a while...a14:=primorial(3358)*primorial(53)*13082761331670030*510510*210*210*30*1296*128.a14 has 13559 digits. The number a14 has a large  sigma_1 valuerelative to itself:  sigma(a14)/(harmonic(a14)+log(harmonic(a14))*exp(harmonic(a14)))  ~=  0.99953340717845609264672369120283054134 .// The expression in 'a14' is related to// the ratio in the Lagarias RH criterion.Cf:"Lagarias discovered an elementary  problem that is equivalent to the [...]"at:< http://en.wikipedia.org/wiki/Jeffrey_Lagarias > .David Bernier-- On Hypnos,http://messagenetcommresearch.com/myths/bios/hypnos.html
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