
Re: abundant numbers, Lagarias criterion for the Riemann Hypothesis
Posted:
Jun 16, 2013 10:34 PM


On 06/16/2013 08:09 AM, David Bernier wrote: > 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 > >
To get high values of Q, one can do better than assigning a nonincreasing monotonic sequence of whole number exponents to the primes 2, 3, 5, 7 ... p_K , the exponents being r_1, r_2, r_3 ... r_K , with j>i ==> r_j <= r_i , r_K = 1 .
The number I mean is: product_{i = 1 ... K} (p_i)^(r_i), with p_i being the i'th prime.

Below, there's a number in a vector, b[10], and I multiply it by the 688 consecutive primes stating at 7187 to reach Q ~= 0.99911099074216 for a number I'll call M.
With either 687 or 689 consecutive primes starting at 7187, the Qvalue is a tiny bit smaller, as shown below.
? 1000000*Qr(b[10]*prod(X=1,689,prime(917+X))) %213 = 999110.98797244567127827686427569616182
? 1000000*Qr(b[10]*prod(X=1,688,prime(917+X))) %214 = 999110.99074216055569607858841451989065
// M := b[10]*prod(X=1,688,prime(917+X))
? 1000000*Qr(b[10]*prod(X=1,687,prime(917+X))) %215 = 999110.99061136341918418963153980094172
The thing that's different is that the sequence of exponents from 2 to the largest prime factor of M is not monotonic. This is so because M is not divisible by any of the primes in an interval [a, b] where a~=6000 and b~= 7000 .
After the current "round" of optimization, I expect primes near 14000 to 15000 (or further) to start making a difference in improving Q. For a reason I don't understand, the primes just above 13567 don't seem to matter for the time being.
On second thought, I'm not sure the resulting sequence of exponents is nonmonotonic: I'd have to check.
What's intriguing is that the best prime to multiply by to get the "optimal" gain in Q shifts around at random.
 On Hypnos, http://messagenetcommresearch.com/myths/bios/hypnos.html

