
Re: abundant numbers, Lagarias criterion for the Riemann Hypothesis
Posted:
Jun 13, 2013 4:45 PM


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 ...
limsup_{n> oo} sigma(n)/( n log(log(n)) ) = exp(gamma), (***) gamma being the EulerMascheroni constant.
This result above, (***), is known as Grönwall's Theorem, dated in the literature to 1913.
I didn't find a reference to a specific article in a journal, a spcific book, or other definite document by Grönwall on the limsup result dated 1913, though.
> > 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

