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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: 3,366
Registered: 12/13/04
Re: abundant numbers, Lagarias criterion for the Riemann Hypothesis
Posted: Jun 19, 2013 8:56 AM
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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 > .

David Bernier

--
On Hypnos,
http://messagenetcommresearch.com/myths/bios/hypnos.html



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