Date: Jun 16, 2013 10:34 PM Author: David Bernier Subject: Re: abundant numbers, Lagarias criterion for the Riemann Hypothesis On 06/16/2013 08:09 AM, David Bernier wrote:

> On 06/14/2013 03:01 PM, David Bernier wrote:

>> On 06/14/2013 01:15 PM, David Bernier wrote:

>>> On 06/14/2013 12:08 PM, James Waldby wrote:

>>>> 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

>>>>> GrĂ¶nwall'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>.

>>>>

>>>

>>>

>>> Ok. so I would go to:

>>> http://wwwhomes.uni-bielefeld.de/achim/highly.html

>>> and from there, download and "gunzip" the file HCNs.gz ...

>>>

>>> Thanks for the feedback.

>>>

>>> I'm looking at finding smallish numbers `n' with unusually high

>>> Q ratio ...

>>>

>>> I get a Q ratio of 0.9925 for some 148-digit number.

>>> Then, 1 - Q is about 0.0075, and I wonder how this relates,

>>> say, to 1/log(n) , so I want a "Gold standard" to

>>> single-out exceptionally abundant numbers ...

>>>

>>>

>>> ? A = 2^10*3^6*5^4*7^3*11^2*13^2*17^2*19^2*23^2;

>>> ? B = 29*31*37*41*43*47*53*59*61*67*71*73*79*83;

>>> ? C = 89*97*101*103*107*109*113*127*131*137*139;

>>> ? D = 149*151*157*163*167*173*179*181*191*193;

>>> ? E = 197*199*211*223*227*229*233*239*241*251;

>>> ? F = 257*263*269*271*277*281*283*293*307*311;

>>> ? G = 313*317*331;

>>> ? n = A*B*C*D*E*F*G; // n is the 148-digit number ...

>>>

>>>

>>> ? harmonic(Z) = Euler+psi(Z+1); // Function definition ...

>>> // Euler = 0.577... and psi is the

>>> digamma function.

>>>

>>>

>>> ? hh = harmonic(n); // sum_{k = 1... n} 1/k using fn. def. above

>>>

>>> ? dd = hh + log(hh)*exp(hh); // Expression based on n'th

>>> // harmonic number

>>>

>>>

>>> ? Q = sigma(n)/dd; // the ratio, Q, for 148-digit `n' gets defined

>>>

>>>

>>> ? Q

>>> %13 = 0.99251022615763635838615903736818502634

>>>

>>>

>>> ? 1 + floor( log(n)/log(10) ) // `n' has 148 digits

>>> %14 = 148

>>

>> I got the HCNs text file.

>>

>> One of the highly composite numbers I'll call `m' ,

>> whose logarithm differs very little from the `n'

>> defined above.

>>

>> We have:

>>

>> sigma(n)/[ exp(gamma)*n*log(log(n)) ] ~= 0.992798

>>

>> and

>>

>> sigma(m)/[ exp(gamma)*m*log(log(m)) ] ~= 0.991642 ,

>>

>> so the 'n' from the epirical work on looking for large

>> Q is somewhat better than the highly-composite 'm'.

>>

>> Actually, m and n have the same number of divisors,

>> (for whatever reason), although 'n' has 67 distinct

>> prime factors, and 'm' has 66 distinct prime factors.

>>

>> n/m = 331/319, and 319 = 11*29.

>>

>> So, n = 331*m/(11*29) , and 331 is prime, just like 11 and 29.

>>

>> The mystery is why, going from 'm' to 'n',

>> switch a factor of 11 and a factor

>> of 29, to be replaced by one prime factor of 331,

>> to get a higher 'Q' ratio?

>

> For a highly composite 273-digit number n defined below

> as a product of three "primorials" together with the small

> integers: 16, 30 and 216 ,

>

> PARI/gp says that:

>

> Q ~= 0.994305962969 .

>

> I'm searching now for a non-trivial multiple of `n' as

> defined below that would have a higher 'Q' ratio,

> but PARI/gp has gone for hours and found nothing.

>

> Wikipedia has a page on the primorial numbers:

> < http://en.wikipedia.org/wiki/Primorial > .

>

> For instance, they say that (p_5)# ,

> "p subscript 5, Sharp Sign",

> denotes the 5th primorial, or 2*3*5*7*11 = 2310.

>

> so,

>

> n is 16*30*216*(p_5)# *(p_11)# * (p_112)# .

>

> By the way, suppose we update the sci.math FAQ,

> what might be said about writing the primorial numbers

> in plain ascii text? The sharp sign, #, makes one of

> its first math notation appearances with that ...

>

> dave

>

>

> ? n =

> 16*30*216*prod(X=1,5,prime(X))*prod(X=1,11,prime(X))*prod(X=1,112,prime(X));

>

> ? hh=Euler+psi(n+1);

> ? sigma(n,1)/(hh+log(hh)*exp(hh))

> %3 = 0.99430596296912309238797312270338679485

> ? floor(log(n)/log(10))+1

> %4 = 273

>

>

To get high values of Q, one can do better than assigning

a non-increasing monotonic sequence of whole number

exponents to the primes 2, 3, 5, 7 ... p_K ,

the exponents being r_1, r_2, r_3 ... r_K ,

with j>i ==> r_j <= r_i , r_K = 1 .

The number I mean is:

product_{i = 1 ... K} (p_i)^(r_i), with p_i being the i'th prime.

---

Below, there's a number in a vector, b[10],

and I multiply it by the 688 consecutive

primes stating at 7187 to reach

Q ~= 0.99911099074216 for a number I'll call M.

With either 687 or 689 consecutive primes

starting at 7187, the Q-value is a tiny bit

smaller, as shown below.

? 1000000*Qr(b[10]*prod(X=1,689,prime(917+X)))

%213 = 999110.98797244567127827686427569616182

? 1000000*Qr(b[10]*prod(X=1,688,prime(917+X)))

%214 = 999110.99074216055569607858841451989065

// M := b[10]*prod(X=1,688,prime(917+X))

? 1000000*Qr(b[10]*prod(X=1,687,prime(917+X)))

%215 = 999110.99061136341918418963153980094172

The thing that's different is that the

sequence of exponents from 2 to the largest

prime factor of M is not monotonic. This is

so because M is not divisible by any of

the primes in an interval [a, b]

where a~=6000 and b~= 7000 .

After the current "round" of optimization,

I expect primes near 14000 to 15000 (or further) to start

making a difference in improving Q. For

a reason I don't understand, the primes just above

13567 don't seem to matter for the time being.

On second thought, I'm not sure the resulting

sequence of exponents is non-monotonic: I'd have

to check.

What's intriguing is that the best prime to

multiply by to get the "optimal" gain in Q

shifts around at random.

--

On Hypnos,

http://messagenetcommresearch.com/myths/bios/hypnos.html