Drexel dragonThe Math ForumDonate to the Math Forum



Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.


Math Forum » Discussions » sci.math.* » sci.math.independent

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

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
David Bernier

Posts: 3,240
Registered: 12/13/04
abundant numbers, Lagarias criterion for the Riemann Hypothesis
Posted: Jun 13, 2013 10:38 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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



Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© Drexel University 1994-2014. All Rights Reserved.
The Math Forum is a research and educational enterprise of the Drexel University School of Education.