en>Fraggle81: Reverted 1 edit by 41.138.86.45 identified using STiki

2014-03-30T15:04:33Z

Reverted 1 edit by 41.138.86.45 identified using STiki

Show changes

en>Décio pinto de jesus at 14:23, 27 February 2014

2014-02-27T14:23:54Z

Show changes

en>Sharma yamijala: Typo

2013-12-18T06:24:03Z

Typo

New page

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

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

where 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 few colossally abundant numbers are [[2 (number)|2]], [[6 (number)|6]], [[12 (number)|12]], [[60 (number)|60]], [[120 (number)|120]], [[360 (number)|360]], 2520, 5040, ... {{OEIS|id=A004490}}; all colossally abundant numbers are also [[superabundant number]]s, but the converse is not true.

==History==

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 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>L. Alaoglu, P. Erdős, "On highly composite and similar numbers", ''Trans. Amer. Math. Soc.'' 56:3 (1944), pp. 448–469, {{MR|0011087}}.</ref>

==Properties==

Colossally abundant numbers are one of several classes of integers 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 π²''n'' / 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''γ''n''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 > 0, the value of the function
:<math>\frac{\sigma(n)}{n^{1+\varepsilon}}</math>
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.<ref>J. C. Lagarias, [http://arxiv.org/abs/math.NT/0008177/ An elementary problem equivalent to the Riemann hypothesis], ''American Mathematical Monthly'' 109 (2002), pp. 534–543.</ref>

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 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 ''pt'' and ''qt'' are [[Rational number|rational]] are the positive integers. Using the corresponding result for three primes—a special case of the [[six exponentials theorem]] that [[Carl Ludwig Siegel|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 factor]]s.

Alaoglu and Erdős's conjecture remains open, although it has been checked up to at least 107.{{Citation needed|date=October 2011}} 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

:<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 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 is true for all ''n'' > 5040:
:<math>\sigma(n)<e^\gamma n \log\log n.\,</math>
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==
{{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]]

Superatom - Revision history

en>Fraggle81: Reverted 1 edit by 41.138.86.45 identified using STiki

en>Décio pinto de jesus at 14:23, 27 February 2014

en>Sharma yamijala: Typo