```Date: Jun 16, 2013 10:34 PM
Author: David Bernier
Subject: Re: abundant numbers, Lagarias criterion for the Riemann Hypothesis

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 Euler-Mascheroni constant.>>>>>>>>>> This result above, (***), is known as>>>>> GrĂ¶nwall'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.uni-bielefeld.de/achim/HCNs.gz>, is shown at>>>> the end of <http://wwwhomes.uni-bielefeld.de/achim/highly.html>.>>>>>>>>>>>>> Ok.  so I would go to:>>> http://wwwhomes.uni-bielefeld.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 148-digit 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>>> single-out 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 148-digit 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 148-digit `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 highly-composite '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 273-digit 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 non-trivial 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 assigninga non-increasing monotonic sequence of whole numberexponents 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 consecutiveprimes stating at 7187 to reachQ ~= 0.99911099074216  for a number I'll call M.With either 687 or 689 consecutive primesstarting at 7187, the Q-value is a tiny bitsmaller, 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 = 999110.99061136341918418963153980094172The thing that's different is that thesequence of exponents from 2 to the largestprime factor of M is not monotonic. This isso because M is not divisible by any ofthe 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 startmaking a difference in improving Q.  Fora reason I don't understand,  the primes just above13567 don't seem to matter for the time being.On second thought, I'm not sure the resultingsequence of exponents is non-monotonic: I'd haveto check.What's intriguing is that the best prime tomultiply by to get the "optimal" gain in Qshifts around at random.-- On Hypnos,http://messagenetcommresearch.com/myths/bios/hypnos.html
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