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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



