Robin's theorem

One of my favorite theorems is Robin's theorem. Consider the function σ(n)/n, where σ(n) is the sum of the divisors of n; I've talked about this function before, when I wrote about the density of abundant numbers. In that post I asked about the distribution of σ(n)/n, so for example we can say that in the limit, σ(n)/n is greater than 2 slightly under a quarter of the time. But another question is just how large can it get?

Gronwall's theorem says that the lim sup of σ(n)/(n log log n) is exp(γ), where γ is Euler's constant. So this function doesn't get very large for "reasonably sized" n, since log log n grows quite slowly, and exp(γ) = 1.781... is not a very large constant. For example, log log 10²⁴ = 4.012... and so σ(10²⁴)/10²⁴ can't be much larger than seven. (One has to be careful with these sorts of limiting results in number theory, though; the Erdos-Kac theorem is another one of my favorite theorems, and it states that the number of distinct prime factors of numbers up to n approaches a normal distribution with mean and variance log log n. But in order to get anything that really "looks like" a normal distribution you probably need log log n to be, say, 10 or so; that limiting distribution is approached very slowly.

Gronwall's theorem is not quite so poor at approaching the limit, assuming the Riemann hypothesis. Robin's theorem is as follows: if the Riemann hypothesis is true, there is a finite list of exceptions, to the inequality

σ(n)/n ≥ exp(γ) log log n

and the largest one is 5040; the full set of exceptions is 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, and 5040. Furthermore, Robin proved in the same paper that if the Riemann hypothesis is false, there are infinitely many counterexamples.

A very similar problem is An elementary problem equivalent to the Riemann hypothesis , which has the virtue of being written about in English and recently enough that it's online. (Robin's theorem is twenty-three years old and in French.) Lagarias shows that

for all n if and only if the Riemann hypothesis is true, where H_n = 1 + 1/2 + ... + 1/n.

You might suspect these are equivalent, since H_n is about (log n) + γ thus exp(H_n) is about n exp(γ) and log(H_n) is about log log n. It's not obvious, though, but it's also not that hard to prove. Lagarias does it in his paper.

Unfortunately, this doesn't make proving the Riemann hypothesis all that much easier. It makes disproving it easier "in theory"; a number which violates Robin's theorem or Lagarias's very similar result constitutes a counterexample to the Riemann hypothesis, and furthermore, if Riemann was wrong then there are infinitely many of them! But pretty much everyone believes the Riemann hypothesis to be true.

And another question, which I've barely thought about; pick a constant C < exp(γ). Is the number of positive integers such that σ(n)/n ≥ C log log n finite? (In other words, are there a lot of "near-counterexamples" to Robin's theorem?)

References:
J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis. Amer. Math. Monthly 109 (2002), 534--543. (arxiv.org/math.NT/0008177)
G. Robin, "Grandes Valeurs de la fonction somme des diviseurs et hypothèse de Riemann." J. Math. Pures Appl. 63, 187-213, 1984.