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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: 3,755 Registered: 12/13/04
Re: abundant numbers, Lagarias criterion for the Riemann Hypothesis
Posted: Jun 21, 2013 2:38 PM

On 06/19/2013 08:56 AM, 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:
>>>
>>> whenener 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 ...
>>
>> 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

>
> I've used PARI/gp to find whole numbers with as large
> a "quotient of abundance" Q as possible, and it has
> taken a while...
>
> a14:=
>
> primorial(3358)*primorial(53)*13082761331670030*510510*210*210*30*1296*128.
>
>
> a14 has 13559 digits. The number a14 has a large sigma_1 value
> relative to itself:
>
> sigma(a14)/(harmonic(a14)+log(harmonic(a14))*exp(harmonic(a14)))
>
>
> ~= 0.99953340717845609264672369120283054134 .
>
> // The expression in 'a14' is related to
> // the ratio in the Lagarias RH criterion.
>
> Cf:
>
> "Lagarias discovered an elementary
> problem that is equivalent to the [...]"
>
> at:
>
> < http://en.wikipedia.org/wiki/Jeffrey_Lagarias > .

Update after more experimentation:

a30 = primorial(8555)*primorial(66)*primorial(16) [continued]
*primorial(8)*primorial(5)*primorial(4) [continued]
*primorial(3)^2*primorial(2)^4*2^8;

Qr(a30) ~= 0.9997306665 .

Qr(W) := sigma(W)/(harmonic(W)+log(harmonic(W))*exp(harmonic(W)))

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