
Re: abundant numbers, Lagarias criterion for the Riemann Hypothesis
Posted:
Jun 14, 2013 12:08 PM


On Thu, 13 Jun 2013 16:45:10 0400, 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: >>> whenever 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. ... >> 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
Q values (with H_n approximated by gamma + log(n+0.5))) for "colossally abundant numbers" <http://oeis.org/A004490> where Q exceeds 0.958 include the following. k Q(a_k) a_k sigma(a_k) Exponents of prime factors of a_k 13. 0.958875 367567200 1889879040 [5, 3, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0] 14. 0.965887 6983776800 37797580800 [5, 3, 2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0] 15. 0.968911 160626866400 907141939200 [5, 3, 2, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0] 16. 0.968922 321253732800 1828682956800 [6, 3, 2, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0] 17. 0.967932 9316358251200 54860488704000 [6, 3, 2, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0] 18. 0.968838 288807105787200 1755535638528000 [6, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0] 19. 0.970746 2021649740510400 12508191424512000 [6, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 0, 0] 20. 0.970641 6064949221531200 37837279059148800 [6, 4, 2, 2, 1, 1, 1, 1, 1, 1, 1, 0, 0] 21. 0.971747 224403121196654400 1437816604247654400 [6, 4, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 0]
Similar results arise for "superior highly composite numbers" <http://en.wikipedia.org/wiki/Superior_highly_composite_number> and <http://oeis.org/A002201>.
I haven't computed Q values for other numbers than the SHCN's and CAN's shown in OEIS. However, it would be quite easy to compute Q values for the first 124260 HCN's, because a 2MB compressed file of them is available which contains ln h_k and ln(sigma(h_k)) at the front of line k, followed by a list of exponents of h_k's prime factorization. The link to the file, <http://wwwhomes.unibielefeld.de/achim/HCNs.gz>, is shown at the end of <http://wwwhomes.unibielefeld.de/achim/highly.html>.
 jiw

