View source for Colossally abundant number

{{Short description|Type of natural number}}
[[Image:Sigma function.svg|thumb|right|[[Divisor function|Sigma function]] {{math|''σ''<sub>1</sub>(''n'')}} up to {{math|1=''n'' = 250}}]]
[[Image:Prime powers in SHCN, CAN.svg|thumb|right|[[Prime power|Prime-power]] factors]]

In [[number theory]], a '''colossally abundant number''' (sometimes abbreviated as '''CA''') is a [[natural number]] that, in a particular, [[Rigour|rigorous]] sense, has many [[divisor]]s. Particularly, it is defined by a [[ratio]] between the [[Aliquot sum|sum of an integer's divisors]] and that integer raised to a [[Exponentiation|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 {{mvar|n}} is said to be colossally abundant if there is an {{math|''ε'' > 0}} such that for all {{math|''k'' > 1}},

:<math>\frac{\sigma(n)}{n^{1+\varepsilon}}\geq\frac{\sigma(k)}{k^{1+\varepsilon}}</math>

where {{mvar|σ}} denotes the [[Divisor function|sum-of-divisors function]].<ref>K. Briggs, "Abundant Numbers and the Riemann Hypothesis", ''Experimental Mathematics'' 15:2 (2006), pp. 251–256, {{doi|10.1080/10586458.2006.10128957}}.</ref>

The first 15 colossally abundant numbers, [[2 (number)|2]], [[6 (number)|6]], [[12 (number)|12]], [[60 (number)|60]], [[120 (number)|120]], [[360 (number)|360]], [[2520 (number)|2520]], [[5040 (number)|5040]], 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800 {{OEIS|id=A004490}} are also the first 15 [[superior highly composite number]]s, but neither set is a subset of the other.

==History==
{{Euler_diagram_numbers_with_many_divisors.svg}}
Colossally abundant numbers were first studied by [[Srinivasa Ramanujan|Ramanujan]] and his findings were intended to be included in his 1915 paper on [[highly composite number]]s.<ref>S. Ramanujan, "Highly Composite Numbers", ''Proc. London Math. Soc.'' 14 (1915), pp. 347–407, {{MR|2280858}}.</ref>  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.<ref>S. Ramanujan, ''Collected papers'', Chelsea, 1962.</ref>  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 [[mathematical proof|proved]] that what would come to be known as [[Robin's inequality]] (see below) holds for all [[sufficiently large]] values of ''n''.<ref>S. Ramanujan, "Highly composite numbers. Annotated and with a foreword by J.-L. Nicholas
and G. Robin", ''Ramanujan Journal'' 1 (1997), pp. 119–153.</ref>

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.<ref name=Alaoglu>{{citation
 | last1 = Alaoglu | first1 = L. | author1-link = Leonidas Alaoglu
 | last2 = Erdős | first2 = P. | author2-link = Paul Erdős
 | journal = Transactions of the American Mathematical Society
 | mr = 0011087
 | pages = 448–469
 | title = On highly composite and similar numbers
 | url = http://www.renyi.hu/~p_erdos/1944-03.pdf
 | volume = 56
 | year = 1944 | issue = 3 | doi=10.2307/1990319| jstor = 1990319 }}.</ref>

==Properties==

Colossally abundant numbers are one of several classes of [[integer]]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 Gustav Heinrich Bachmann|Paul Bachmann]] showed that on average, σ(''n'') is around π{{sup|2}}''n''&thinsp;/&thinsp;6.<ref name=HW>G. Hardy, E. M. Wright, ''An Introduction to the Theory of Numbers. Fifth Edition'', Oxford Univ. Press, Oxford, 1979.</ref> [[Thomas Hakon Grönwall|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''<sup>γ</sup>''n''&thinsp;log(log(''n'')), where γ is the [[Euler–Mascheroni constant]].<ref name=HW />  Hence colossally abundant numbers capture the notion of having many divisors by requiring them to maximise, for some ε&nbsp;>&nbsp;0, the value of the [[function (mathematics)|function]]
:<math>\frac{\sigma(n)}{n^{1+\varepsilon}}</math>
over all values of ''n''.  Bachmann and Grönwall's results ensure that for every ε&nbsp;>&nbsp;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<sup>18</sup>.<ref name=Lagarias>J. C. Lagarias, [https://arxiv.org/abs/math.NT/0008177/ An elementary problem equivalent to the Riemann hypothesis], ''American Mathematical Monthly'' 109 (2002), pp. 534–543.</ref>

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]]s. Let

:<math>e_p(\varepsilon) = \left\lfloor\frac{\ln\left(1+\displaystyle\frac{p-1}{p^{1+\varepsilon}-p}\right)}{\ln p}\right\rfloor\quad</math>

for any [[prime number]] ''p'' and positive real <math>\varepsilon</math>. Then

:<math>\quad s(\varepsilon) = \prod_{p \in \mathbb{P}} p^{e_p(\varepsilon)}\ </math> 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.<ref>P. Erdős, J.-L. Nicolas, "Répartition des nombres superabondants", ''Bull. Math. Soc. France'' 103 (1975), pp. 65–90.</ref>

In their 1944 paper, Alaoglu and Erdős [[conjecture]]d 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<sup>t</sup>'' and ''q<sup>t</sup>'' are [[Rational number|rational]] are the positive integers.  Using the corresponding result for three primes, which [[Carl Ludwig Siegel|Siegel]] assured them was true<ref>Alaoglu and Erdős, (1944), p.455: "Professor Siegel has communicated to us the result that ''q''<sup>&nbsp;''x''</sup>, ''r''<sup>&nbsp;''x''</sup> and ''s''<sup>&nbsp;''x''</sup> can not be simultaneously rational except if ''x'' is an integer."</ref>—a special case of the [[six exponentials theorem]] proven in the 1960s<ref name="Waldschmidt 2022 pp. 599–607">{{cite journal | last=Waldschmidt | first=Michel | title=Six Exponentials Theorem — Irrationality | journal=Resonance | volume=27 | issue=4 | date=2022 | issn=0973-712X | doi=10.1007/s12045-022-1351-0 | pages=599–607| s2cid=248307621 }}</ref> by [[Serge Lang]] and [[Kanakanahalli Ramachandra|K. Ramachandra]] —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 factor]]s). The quotient can never be the [[square (algebra)|square]] of a prime.

Alaoglu and Erdős's conjecture remains open, although it has been checked up to at least 10<sup>7</sup>.<ref>{{Cite OEIS|A073751|name=Prime numbers that when multiplied in order yield the sequence of colossally abundant numbers}}</ref>  If true it would mean that there was a sequence of non-distinct prime numbers ''p''<sub>1</sub>, ''p''<sub>2</sub>, ''p''<sub>3</sub>,... such that the ''n''th colossally abundant number was of the form

:<math>c_n = \prod_{i=1}^n p_{i}</math>

Assuming the conjecture holds, this sequence of primes begins 2, 3, 2, 5, 2, 3, 7, 2 {{OEIS|id=A073751}}.  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 number]]s, colossally abundant numbers are a generalization of [[Abundant number|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.<ref>{{cite OEIS|A004490|Colossally abundant numbers}} "A subsequence of A004394 (superabundant numbers)."</ref>

== Relation to the Riemann hypothesis ==
In the 1980s Guy Robin showed<ref>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.</ref> that the [[Riemann hypothesis]] is equivalent to the assertion that the following [[inequality (mathematics)|inequality]] is true for all ''n''&nbsp;>&nbsp;5040: (where γ is the [[Euler–Mascheroni constant]])
:<math>\sigma(n)<e^\gamma n \log\log n \approx 1.781072418 n \log\log n \,</math>
This inequality is known to fail for 27 numbers {{OEIS|id=A067698}}:
: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''&nbsp;=&nbsp;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''&nbsp;>&nbsp;5040.

In 2001–2 Lagarias<ref name=Lagarias/> demonstrated an alternate form of Robin's assertion which requires no exceptions, using the [[harmonic number]]s instead of log:
:<math>\sigma(n) < H_n + \exp(H_n)\log(H_n)</math>

Or, other than the 8 exceptions of ''n'' = 1, 2, 3, 4, 6, 12, 24, 60:
:<math>\sigma(n) < \exp(H_n)\log(H_n)</math>

==References==
{{reflist}}

==External links==
*[http://keithbriggs.info/abundant.html Keith Briggs on colossally abundant numbers and the Riemann hypothesis]
*[http://mathworld.wolfram.com/ColossallyAbundantNumber.html MathWorld entry]
*[http://keithbriggs.info/documents/RH_abundant-pp.pdf Notes on the Riemann hypothesis and abundant numbers]
*[http://www.mpim-bonn.mpg.de/preprints/send?year=&number=&name=&title=robin More on Robin's formulation of the RH]

{{Divisor classes}}
{{Classes of natural numbers}}

{{DEFAULTSORT:Colossally Abundant Number}}
[[Category:Divisor function]]
[[Category:Integer sequences]]