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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 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 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
>
>

To get high values of Q, one can do better than assigning
a non-increasing 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 Q-value 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 non-monotonic: 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

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