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Topic: abundant numbers, Lagarias criterion for the Riemann Hypothesis
Replies: 13   Last Post: Jul 18, 2013 2:13 AM

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 David Bernier Posts: 3,892 Registered: 12/13/04
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 Euler-Mascheroni 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.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

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

David Bernier

--
On Hypnos,
http://messagenetcommresearch.com/myths/bios/hypnos.html

Date Subject Author
6/13/13 David Bernier
6/13/13 David Bernier
6/13/13 David Bernier
6/13/13 Brian Q. Hutchings
6/19/13 David Bernier
6/21/13 David Bernier
7/18/13 David Bernier
6/14/13 James Waldby
6/14/13 David Bernier
6/14/13 David Bernier
6/14/13 Brian Q. Hutchings
6/16/13 David Bernier
6/16/13 David Bernier