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Topic: abundant numbers, Lagarias criterion for the Riemann Hypothesis
Replies: 13   Last Post: Jul 18, 2013 2:13 AM

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David Bernier

Posts: 3,190
Registered: 12/13/04
Re: abundant numbers, Lagarias criterion for the Riemann Hypothesis
Posted: Jun 13, 2013 12:52 PM
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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

>
> David Bernier



--
On Hypnos,
http://messagenetcommresearch.com/myths/bios/hypnos.html



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