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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 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 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

>
> 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

--
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