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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: 3,672 Registered: 12/13/04
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

Date Subject Author
6/13/13 David Bernier
6/13/13 David Bernier
6/13/13 David Bernier
6/13/13 Brian Q. Hutchings
6/19/13 David Bernier
6/21/13 David Bernier
7/18/13 David Bernier
6/14/13 James Waldby
6/14/13 David Bernier
6/14/13 David Bernier
6/14/13 Brian Q. Hutchings
6/16/13 David Bernier
6/16/13 David Bernier