```Date: Jun 21, 2013 2:38 PM
Author: David Bernier
Subject: Re: abundant numbers, Lagarias criterion for the Riemann Hypothesis

On 06/19/2013 08:56 AM, David Bernier wrote:> 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) = 12,508,191,424,512,000>> I've used PARI/gp to find whole numbers with as large> a "quotient of abundance" Q as possible, and it has> taken 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 value> relative 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 > .Update after more experimentation:a30 = primorial(8555)*primorial(66)*primorial(16) [continued]       *primorial(8)*primorial(5)*primorial(4)     [continued]       *primorial(3)^2*primorial(2)^4*2^8;Qr(a30) ~= 0.9997306665 .Qr(W) := sigma(W)/(harmonic(W)+log(harmonic(W))*exp(harmonic(W)))David Bernier-- On Hypnos,http://messagenetcommresearch.com/myths/bios/hypnos.html
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