
Re: abundant numbers, Lagarias criterion for the Riemann Hypothesis
Posted:
Jun 14, 2013 3:01 PM


On 06/14/2013 01:15 PM, David Bernier wrote: > 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
I got the HCNs text file.
One of the highly composite numbers I'll call `m' , whose logarithm differs very little from the `n' defined above.
We have:
sigma(n)/[ exp(gamma)*n*log(log(n)) ] ~= 0.992798
and
sigma(m)/[ exp(gamma)*m*log(log(m)) ] ~= 0.991642 ,
so the 'n' from the epirical work on looking for large Q is somewhat better than the highlycomposite 'm'.
Actually, m and n have the same number of divisors, (for whatever reason), although 'n' has 67 distinct prime factors, and 'm' has 66 distinct prime factors.
n/m = 331/319, and 319 = 11*29.
So, n = 331*m/(11*29) , and 331 is prime, just like 11 and 29.
The mystery is why, going from 'm' to 'n', switch a factor of 11 and a factor of 29, to be replaced by one prime factor of 331, to get a higher 'Q' ratio?
The first five primes after 300 are 307, 311, 313, 317 and 331 .
They are all factors of 'n'.
So, 307, 311, 313 and 317 are factors of the highly composite number 'm', but not 331.
Instead, 'm' can be obtained as: (n/331)*11*29 .
David Bernier
Work:
? (sigma(n, 1)/(n*log(log(n))))/exp(Euler) %53 = 0.99279892618630412257048797008815184714
? (sigma(m, 1)/(m*log(log(m))))/exp(Euler) %54 = 0.99164208754504011740275423920309459284
? n/m %55 = 331/319
 On Hypnos, http://messagenetcommresearch.com/myths/bios/hypnos.html

