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Computations on 8000 colossally abundant numbers out to exp(exp(29.710462))

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Variation of log(delta(n)) from Best-fit linear function of log(log(n)) Variation of log(delta(n)) from Best-fit linear function of log(log(n))

Reminder:  For a colossally abundant number n, with n > 5040, one can define

delta(n) := exp(gamma)*log(log(n)) – sigma(n)/n               [Briggs, def. 4-2, page 254 ] .

This is as in the paper by Keith Briggs,

Briggs, K.,  2006, “Abundant Numbers and the Riemann Hypothesis“.

sigma(.) is the sum of divisors function, sigma(n)/n is the same as the sum of the inverses of all divisors of n [easy proof], also known as the “abundancy index of n”;  gamma is the Euler-Mascheroni constant 0.57721 …

The Theorem of Guy Robin is that the Riemann Hypothesis is equivalent to :

exp(gamma)*log(log(n)) – sigma(n)/n > 0 , for all  n > 5040. Also see Grönwall’s theorem on the asymptotic growth rate of the sigma function, at Wikipedia:

https://en.wikipedia.org/wiki/Divisor_function#Approximate_growth_rate  .

We can plot how delta(n) changes with…

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Written by meditationatae

July 17, 2015 at 12:28 pm

Posted in History

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