Colossally abundant number

Formally, a number n is said to be colossally abundant if there is an ε > 0 such that for all k > 1, where σ denotes the sum-of-divisors function.

[3] His findings were mostly conditional on the Riemann hypothesis and with this assumption he found upper and lower bounds for the size of colossally abundant numbers and proved that what would come to be known as Robin's inequality (see below) holds for all sufficiently large values of n.[4] The class of numbers was reconsidered in a slightly stronger form in a 1944 paper of Leonidas Alaoglu and Paul Erdős in which they tried to extend Ramanujan's results.

For a positive integer n, the sum-of-divisors function σ(n) gives the sum of all those numbers that divide n, including 1 and n itself.

[6] Grönwall's theorem, meanwhile, says that the maximal order of σ(n) is ever so slightly larger, specifically there is an increasing sequence of integers n such that for these integers σ(n) is roughly the same size as eγn log(log(n)), where γ is the Euler–Mascheroni constant.

Alaoglu and Erdős's conjecture would also mean that no value of ε gives four different integers n as maxima of the above function.

[12] In the 1980s Guy Robin showed[13] that the Riemann hypothesis is equivalent to the assertion that the following inequality is true for all n > 5040: (where γ is the Euler–Mascheroni constant) This inequality is known to fail for 27 numbers (sequence A067698 in the OEIS): Robin showed that if the Riemann hypothesis is true then n = 5040 is the last integer for which it fails.

It is known that Robin's inequality, if it ever fails to hold, will fail for a colossally abundant number n; thus the Riemann hypothesis is in fact equivalent to Robin's inequality holding for every colossally abundant number n > 5040.