
Re: abundant numbers, Lagarias criterion for the Riemann Hypothesis
Posted:
Jun 16, 2013 8:09 AM


On 06/14/2013 03:01 PM, David Bernier wrote: > 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?
For a highly composite 273digit number n defined below as a product of three "primorials" together with the small integers: 16, 30 and 216 ,
PARI/gp says that:
Q ~= 0.994305962969 .
I'm searching now for a nontrivial multiple of `n' as defined below that would have a higher 'Q' ratio, but PARI/gp has gone for hours and found nothing.
Wikipedia has a page on the primorial numbers: < http://en.wikipedia.org/wiki/Primorial > .
For instance, they say that (p_5)# , "p subscript 5, Sharp Sign", denotes the 5th primorial, or 2*3*5*7*11 = 2310.
so,
n is 16*30*216*(p_5)# *(p_11)# * (p_112)# .
By the way, suppose we update the sci.math FAQ, what might be said about writing the primorial numbers in plain ascii text? The sharp sign, #, makes one of its first math notation appearances with that ...
dave
? n = 16*30*216*prod(X=1,5,prime(X))*prod(X=1,11,prime(X))*prod(X=1,112,prime(X)); ? hh=Euler+psi(n+1); ? sigma(n,1)/(hh+log(hh)*exp(hh)) %3 = 0.99430596296912309238797312270338679485 ? floor(log(n)/log(10))+1 %4 = 273
 On Hypnos, http://messagenetcommresearch.com/myths/bios/hypnos.html

