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In mathematics, a colossally abundant number (sometimes abbreviated as CA) is a natural number that, in a particular rigorous sense, has many divisors. Formally, a number n is colossally abundant if and only if there is an ε > 0 such that for all k > 1,

\( \frac{\sigma(n)}{n^{1+\varepsilon}}\geq\frac{\sigma(k)}{k^{1+\varepsilon}} \)

where σ denotes the sum-of-divisors function.[1] The first few colossally abundant numbers are 2, 6, 12, 60, 120, 360, 2520, 5040, ... (sequence A004490 in OEIS); all colossally abundant numbers are also superabundant numbers, but the converse is not true.

History

Colossally abundant numbers were first studied by Ramanujan and his findings were intended to be included in his 1915 paper on highly composite numbers.[2] Unfortunately, the publisher of the journal to which Ramanujan submitted his work, the London Mathematical Society, was in financial difficulties at the time and Ramanujan agreed to remove aspects of the work in order to reduce the cost of printing.[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.[5]
Properties

Colossally abundant numbers are one of several classes of integers that try to capture the notion of having lots of divisors. 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. Paul Bachmann showed that on average, σ(n) is around π²n / 6.[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 e^γnlog(log(n)), where γ is the Euler–Mascheroni constant.[6] Hence colossally abundant numbers capture the notion of having lots of divisors by requiring them to maximise, for some ε > 0, the value of the function

\( \frac{\sigma(n)}{n^{1+\varepsilon}} \)

over all values of n. Bachmann and Grönwall's results ensure that for every ε > 0 this function has a maximum and that as ε tends to zero these maxima will increase. Thus there are infinitely many colossally abundant numbers, although they are rather sparse, with only 22 of them less than 1018.[7]

For every ε the above function has a maximum, but it is not obvious, and in fact not true, that for every ε this maximum value is unique. Alaoglu and Erdős studied how many different values of n could give the same maximal value of the above function for a given value of ε. They showed that for most values of ε there would be a single integer n maximising the function. Later, however, Erdős and Jean-Louis Nicolas showed that for a certain set of discrete values of ε there could be two or four different values of n giving the same maximal value.[8]

In their 1944 paper, Alaoglu and Erdős managed to show that the ratio of two consecutive superabundant numbers was always a prime number, but couldn't quite achieve this for colossally abundant numbers. They did conjecture that this was the case and showed that it would follow from a special case of the four exponentials conjecture in transcendental number theory, specifically that for any two distinct prime numbers p and q, the only real numbers t for which both pt and qt are rational are the positive integers. Using the corresponding result for three primes—a special case of the six exponentials theorem that Siegel claimed to have proven—they managed to show that the quotient of two consecutive colossally abundant numbers is always either a prime or a semiprime, that is a number with just two prime factors.

Alaoglu and Erdős's conjecture remains open, although it has been checked up to at least 107. If true it would mean that there was a sequence of non-distinct prime numbers p₁, p₂, p₃,… such that the nth colossally abundant number was of the form

\( c_n=p_1p_2\cdots p_n.\, \)

Assuming the conjecture holds, this sequence of primes begins 2, 3, 2, 5, 2, 3, 7, 2 (sequence A073751 in OEIS). Alaoglu and Erdős's conjecture would also mean that no value of ε gives four different integers n as maxima of the above function.
Relation to the Riemann hypothesis

In the 1980s Guy Robin showed[9] that the Riemann hypothesis is equivalent to the assertion that the following inequality is true for all n > 5040:

\( \sigma(n)<e^\gamma n \log\log n.\, \)

This inequality is known to fail at n = 5040, but Robin showed that if the Riemann hypothesis is true then this is the last integer for which it fails. The inequality is now known as Robin's inequality after his work. 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.
References

^ K. Briggs, "Abundant Numbers and the Riemann Hypothesis", Experimental Mathematics 15:2 (2006), pp. 251–256, doi:10.1080/10586458.2006.10128957 .
^ S. Ramanujan, "Highly Composite Numbers", Proc. London Math. Soc. 14 (1915), pp. 347–407, MR2280858.
^ S. Ramanujan, Collected papers, Chelsea, 1962.
^ S. Ramanujan, "Highly composite numbers. Annotated and with a foreword by J.-L. Nicholas and G. Robin", Ramanujan Journal 1 (1997), pp. 119–153.
^ L. Alaoglu, P. Erdős, "On highly composite and similar numbers", Trans. Amer. Math. Soc. 56:3 (1944), pp. 448–469, MR0011087.
^ a b G. Hardy, E. M. Wright, An Introduction to the Theory of Numbers. Fifth Edition, Oxford Univ. Press, Oxford, 1979.
^ J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, American Mathematical Monthly 109 (2002), pp. 534–543.
^ P. Erdős, J.-L. Nicolas, "Répartition des nombres superabondants", Bull. Math. Soc. France 103 (1975), pp. 65–90.
^ G. Robin, "Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann", Journal de Mathématiques Pures et Appliquées 63 (1984), pp. 187-213.

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