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Topic:
abundant numbers, Lagarias criterion for the Riemann Hypothesis
Replies:
13
Last Post:
Jul 18, 2013 2:13 AM




abundant numbers, Lagarias criterion for the Riemann Hypothesis
Posted:
Jun 13, 2013 10:38 AM


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) ] ?
David Bernier  On Hypnos, http://messagenetcommresearch.com/myths/bios/hypnos.html



