Colossally abundant number

In number theory, a colossally abundant number (sometimes abbreviated as CA) is a natural number that, in a particular, rigorous sense, has many divisors. Particularly, it is defined by a ratio between the sum of an integer's divisors and that integer raised to a power higher than one. For any such exponent, whichever integer has the highest ratio is a colossally abundant number. It is a stronger restriction than that of a superabundant number, but not strictly stronger than that of an abundant number.

Formally, a number n is said to be colossally abundant if there is an ε > 0 such that for all k > 1,

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

where σ denotes the sum-of-divisors function.^[1]

The first 15 colossally abundant numbers,

Colossally abundant numbers were first studied by

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

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]

Just like with superior highly composite numbers, an effective construction of the set of all colossally abundant numbers is given by the following monotonic mapping from the positive real numbers. Let

${\displaystyle e_{p}(\varepsilon )=\left\lfloor {\frac {\ln \left(1+\displaystyle {\frac {p-1}{p^{1+\varepsilon }-p}}\right)}{\ln p}}\right\rfloor \quad }$

for any prime number p and positive real ${\displaystyle \varepsilon }$. Then

${\displaystyle \quad s(\varepsilon )=\prod _{p\in \mathbb {P} }p^{e_{p}(\varepsilon )}\ }$ is a colossally abundant number.

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

Alaoglu and Erdős's conjecture remains open, although it has been checked up to at least 10⁷.[11] 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

${\displaystyle c_{n}=\prod _{i=1}^{n}p_{i}}$

Assuming the conjecture holds, this sequence of primes begins 2, 3, 2, 5, 2, 3, 7, 2 (sequence A073751 in the 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 abundant numbers

Like superabundant numbers, colossally abundant numbers are a generalization of abundant numbers. Also like superabundant numbers, it is not a strict generalization; a number can be colossally abundant without being abundant. This is true in the case of 6; 6's divisors are 1,2,3, and 6, but an abundant number is defined to be one where the sum of the divisors, excluding itself, is greater than the number itself; 1+2+3=6, so this condition is not met (and 6 is instead a perfect number). However all colossally abundant numbers are also superabundant numbers.^[12]

Relation to the Riemann hypothesis

In the 1980s Guy Robin showed

${\displaystyle \sigma (n)<e^{\gamma }n\log \log n\approx 1.781072418n\log \log n\,}$

This inequality is known to fail for 27 numbers (sequence A067698 in the OEIS):

2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 18, 20, 24, 30, 36, 48, 60, 72, 84, 120, 180, 240, 360, 720, 840, 2520, 5040

Robin showed that if the Riemann hypothesis is true then n = 5040 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.

In 2001–2 Lagarias^[7] demonstrated an alternate form of Robin's assertion which requires no exceptions, using the harmonic numbers instead of log:

${\displaystyle \sigma (n)<H_{n}+\exp(H_{n})\log(H_{n})}$

Or, other than the 8 exceptions of n = 1, 2, 3, 4, 6, 12, 24, 60:

${\displaystyle \sigma (n)<\exp(H_{n})\log(H_{n})}$

References