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




Re: abundant numbers, Lagarias criterion for the Riemann Hypothesis
Posted:
Jun 13, 2013 12:52 PM


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



