
Re: abundant numbers, Lagarias criterion for the Riemann Hypothesis
Posted:
Jun 14, 2013 1:15 PM


On 06/14/2013 12:08 PM, James Waldby wrote: > 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>. >
Ok. so I would go to: http://wwwhomes.unibielefeld.de/achim/highly.html and from there, download and "gunzip" the file HCNs.gz ...
Thanks for the feedback.
I'm looking at finding smallish numbers `n' with unusually high Q ratio ...
I get a Q ratio of 0.9925 for some 148digit number. Then, 1  Q is about 0.0075, and I wonder how this relates, say, to 1/log(n) , so I want a "Gold standard" to singleout exceptionally abundant numbers ...
? A = 2^10*3^6*5^4*7^3*11^2*13^2*17^2*19^2*23^2; ? B = 29*31*37*41*43*47*53*59*61*67*71*73*79*83; ? C = 89*97*101*103*107*109*113*127*131*137*139; ? D = 149*151*157*163*167*173*179*181*191*193; ? E = 197*199*211*223*227*229*233*239*241*251; ? F = 257*263*269*271*277*281*283*293*307*311; ? G = 313*317*331; ? n = A*B*C*D*E*F*G; // n is the 148digit number ...
? harmonic(Z) = Euler+psi(Z+1); // Function definition ... // Euler = 0.577... and psi is the digamma function.
? hh = harmonic(n); // sum_{k = 1... n} 1/k using fn. def. above
? dd = hh + log(hh)*exp(hh); // Expression based on n'th // harmonic number
? Q = sigma(n)/dd; // the ratio, Q, for 148digit `n' gets defined
? Q %13 = 0.99251022615763635838615903736818502634
? 1 + floor( log(n)/log(10) ) // `n' has 148 digits %14 = 148
David Bernier
 On Hypnos, http://messagenetcommresearch.com/myths/bios/hypnos.html

