In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a colossally abundant number (sometimes abbreviated as CA) is a
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
that, in a particular, rigorous sense, has many
divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
s. 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. All colossally abundant numbers are also
superabundant numbers, but the
converse is not true.
The first 15 colossally abundant numbers,
2,
6,
12,
60,
120,
360,
2520,
5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800 are also the first 15
superior highly composite numbers, but neither set is a subset of the other.
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. 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 to reduce the cost of printing. 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
In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as ''the'' divisor function, it counts the ''number of divisors of an integer'' (includin ...
(see below) holds for all
sufficiently large values of ''n''.
The class of numbers was reconsidered in a slightly stronger form in a 1944 paper of
Leonidas Alaoglu
Leonidas (''Leon'') Alaoglu ( el, Λεωνίδας Αλάογλου; March 19, 1914 – August 1981) was a mathematician, known for his result, called Alaoglu's theorem on the weak-star compactness of the closed unit ball in the dual of a n ...
and
Paul Erdős in which they tried to extend Ramanujan's results.
[.]
Properties
Colossally abundant numbers are one of several classes of
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s that try to capture the notion of having many 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 π''n'' / 6.
[G. Hardy, E. M. Wright, ''An Introduction to the Theory of Numbers. Fifth Edition'', Oxford Univ. Press, Oxford, 1979.] 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''
γ''n'' log(log(''n'')), where γ is the
Euler–Mascheroni constant.
Hence colossally abundant numbers capture the notion of having many divisors by requiring them to maximise, for some ε > 0, the value of the
function
:
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 10
18.
[J. C. Lagarias]
An elementary problem equivalent to the Riemann hypothesis
''American Mathematical Monthly'' 109 (2002), pp. 534–543.
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 number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s. Let
:
for any
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
''p'' and positive real
. Then
:
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.
In their 1944 paper, Alaoglu and Erdős
conjectured that the ratio of two consecutive colossally abundant numbers was always a prime number. They showed that this 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 ''p
t'' and ''q
t'' are
rational are the positive integers. Using the corresponding result for three primes—a special case of the
six exponentials theorem that
Siegel
Siegel (also Segal or Segel), is a German and Ashkenazi Jewish surname. it can be traced to 11th century Bavaria and was used by people who made wax seals for or sealed official documents (each such male being described as a ''Siegelbeamter''). Al ...
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
In mathematics, a semiprime is a natural number that is the product of exactly two prime numbers. The two primes in the product may equal each other, so the semiprimes include the squares of prime numbers.
Because there are infinitely many prime ...
(that is, a number with just two
prime factors). The quotient can never be the
square of a prime.
Alaoglu and Erdős's conjecture remains open, although it has been checked up to at least 10
7. If true it would mean that there was a sequence of non-distinct prime numbers ''p''
1, ''p''
2, ''p''
3,... such that the ''n''th colossally abundant number was of the form
:
Assuming the conjecture holds, this sequence of primes begins 2, 3, 2, 5, 2, 3, 7, 2 . 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
[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.] 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)
: