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

Updated today …

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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