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) = 12,508,191,424,512,000

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 > .

David Bernier

--

On Hypnos,

http://messagenetcommresearch.com/myths/bios/hypnos.html