Superabundant Numbers and the Riemann Hypothesis
Amir Akbary and Zachary FriggstadPThe function σ(n) = d|n d is the sum of divisors function, so for example σ(12) = 28. In
1913 Gronwall proved thatσ(n)= 1,lim sup γn→∞ e n log log n
where γ ≈ 0.57721 is Euler’s constant (see [4, Theorem 323]). This says that the maximal sizeof σ(n) is roughly eγ n log log n. The following theorem of Robin [7, Theorem 2] gives a morerefined version of this upper bound.
Theorem 1 For n ≥ 3 we haveσ(n)0.6483.

(1)

σ(n)
< 1, for n ≥ 5041.
eγ n log log n(2)eγ n log log n<1+eγ (log log n)2

In [7] Robin also proved the following striking result.
Theorem 2 The Riemann Hypothesis is true if and only if

For a lively exposition of this theorem and its connection to the Riemann Hypothesis see [5]. In
this note, we propose a method that will establish explicit upper bounds for σ(n)/eγ n log log n.
Our main observation is that the least number violating the inequality (2) should be a superabundant number.
A positive integer n is said to be superabundant if σ(m)/m < σ(n)/n for all m < n. The
first 20 superabundant numbers are 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720,840, 1260, 1680, 2520, 5040, 10080. The sequence of superabundant numbers is the sequence
A004394 in Sloane’s Encyclopedia [8]. The list of the first 500 superabundant numbers isavailable at [6], whereM = 25484247877474623694559469201315033045359474150161923076850486576760360768000
is the largest superabundant number in the list.
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Theorem 3 If there is any counterexample to Robin’s inequality (2), then the least such counterexample is a superabundant number.
Proof It is known that there is no counterexample n to Robin’s inequality (2) with 5040 <n ≤ 10080. Note that 10080 is a superabundant number.
Now, assume that n > 10080 is the minimal counterexample to the inequality and also
assume that n is not superabundant. Then there is a k < n such that σ(k)/k ≥ σ(n)/n.
Without loss of generality we may assume that k ≥ 10080. This is true since if k < 10080,then σ(k)/k < σ(10080)/10080 (as 10080 is superabundant) and so we could choose insteadk = 10080.
We next note that sinceσ(k)σ(n)≥≥ eγ log log n > eγ log log kkn
then k (≥ 10080 > 5040) is a counterexample to Robin’s inequality (2). However, we assumed that n was the minimal counterexample, so this is a contradiction. Therefore, if acounterexample > 5040 exists, then the least counterexample must be superabundant. We point out that superabundant numbers form a very thin subset of the natural numbers.
More precisely, let S(x) be the number of superabundant numbers not exceeding x. Then in[1], Alaoglu and Erdös state thatlog (S(x))(log log x)2
is bounded. It is an open problem whether
log (S(x))
log log x(3)is bounded or not. In [2], Briggs reports that the ratio (3) gives about 1.2 at log log x = 5.
Moreover, Erdös and Nicolas [3] proved that
lim infx→∞5log (S(x))≥1+≈ 1.1042.
log log x
48

As a consequence of Theorem 3, one can attempt to disprove the Riemann Hypothesis
computationally by testing the inequality (2) for superabundant numbers. See [2] for information regarding this approach and methods for generating superabundant numbers. The authorof [2] also mentions that, at the moment, there is no (guaranteed, fast) algorithmic method
known for finding all superabundant numbers below a given bound.
Finally we can numerically verify that the only numbers in the list of the first 500 superabundant numbers in [6] that do not satisfy Robin’s inequality (2) are 2, 4, 6, 12, 24, 36, 48,2

60, 120, 180, 240, 360, 720, 840, 2520, 5040. So as a direct corollary of the above theorems wehaveσ(n)eγ n log log n<1+0.6483
eγ (log log M )2

= 1.013617,for n ≥ 5041, where M is the superabundant number recorded above.
ACKNOWLEDGMENT The authors would like to thank Prof. Ram Murty and Prof.
Jeffrey Lagarias for their suggestions. Research of both authors is partially supported byNSERC.

References[1] L. Alaoglu and P. Erdös, On highly composite and similar numbers, Trans. Amer. Math.
Soc. 56 (1944) 448–469.
[2] K. Briggs, Abundant numbers and the Riemann hypothesis, Experiment. Math. 15 (2006)
251–256.
[3] P. Erdös and J.-L. Nicolas, Répartition des nombres superabondants, Bull. Soc. Math.
France 103 (1975) 65–90.
[4] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford
University Press, Oxford, 1979.
[5] J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, thisMonthly 109 (2002) 534–543.
[6] T. D. Noe, Table of first 500 superabundant numbers,
http://www.research.att.com/∼njas/sequences/b004394.txt.

availableat[7] G. Robin, Grandes valeurs de la fonction somme de diviseurs et hypothèse de Riemann,J. Math. Pure Appl. (9) 63 (1984) 187–213.
[8] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, available at
http://www.research.att.com/∼njas/sequences/.
Department of Mathematics and Computer Science, University of Lethbridge, 4401 University Drive West, Lethbridge, Alberta,
T1K 3M4, CANADA
E–mail address: amir.akbary@uleth.ca
Department of Computing Science, University of Alberta, Edmonton, Alberta, T6G 2E8, CANADA
E–mail address: zacharyf@cs.ualberta.ca

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