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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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James Waldby

Posts: 366
Registered: 1/27/11
Re: abundant numbers, Lagarias criterion for the Riemann Hypothesis
Posted: Jun 14, 2013 12:08 PM
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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>.

--
jiw




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