Date: Jun 21, 2013 2:38 PM Author: David Bernier Subject: Re: abundant numbers, Lagarias criterion for the Riemann Hypothesis 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

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