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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: Jul 18, 2013 2:13 AM

On 06/21/2013 02:38 PM, David Bernier wrote:
> 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)))

[...]

Copied output from PARI/gp with comments.

Best: 1 Score: 9999.750775 007375 490174 519248 3295500557
[306738, 418, 50, 18, 9, 6, 5, 4, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1,
1, 1, 1, 1]

----
The number 'n' is a product of 25 primorials represented in the vector:

n := p#(306738)*p#(418) ... * (p#(1))^9.

Q(n):= sigma(n)/[ H_n + log(H_n)*exp(H_n) ] ,

sigma() is the sum of divisors function, H_n is the
n'th harmonic number:
H_n = sum_{k = 1...n} 1/k .

Jeffrey Lagarias showed: RH <==> Q(n) < 1 for all integers n > 2.
Gronwall first worked out the asymptotic extreme behaviour of sigma()
to "first order" .

The score of 'n' is 10000*Q(n).

Try to replace p#(m_j) by p#(m_j +1), j = 1 ... 25 in succession.
The winning 'j' is the one for which the Q-value is the largest.

m_1 = 306738, m_2 = 418, m_3 = 50, ... m_17 = m_18 = ... = m_24 = m_25 = 1 .

Very occasionally, the best 'j' means a _lower_ Q-value:

9999.750775004441 < 9999.750775007375 :
-----

1 9999.750775 004441 457325 932065 3334422049 [ Best is
from using j=1]
Best: 1 Score: 9999.7507750044414573259320653334422049

1 9999.750774 999987 661808 339470 2904857568 [ Best is from
using j=1]
Best: 1 Score: 9999.7507749999876618083394702904857568

1 9999.750775 001859 641379 985230 1801404019 [ Best is from
using j=1]
Best: 1 Score: 9999.7507750018596413799852301801404019

1 9999.750775 004453 408349 110566 0971511393 [ Best is from
using j=1]
Best: 1 Score: 9999.7507750044534083491105660971511393

1 9999.750775 003285 856812 854132 7209233142 [ Best is from
using j=1]
Best: 1 Score: 9999.7507750032858568128541327209233142
[306743, 418, 50, 18, 9, 6, 5, 4, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1,
1, 1, 1, 1]

----
The new m_1 = 306743 = 306738 + 5.

Score 5 ago:
Score: 9999.750775 007375 490174 519248 329550 0557

current score:
Score: 9999.750775 003285 856812 854132 720923 3142

Current score < (Score 5 ago).
----

It's a temporary dip, no doubt ....

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