The Riemann Hypothesis Says 5040 Is the Last

[troymc]

This provides random Joe a way to show that the Riemann Hypothesis is false: try a bunch of big numbers (bigger than 5040) in σ(n)/(n ln(ln (n))) and check if the result is greater than or equal to e^γ (1.7810724179901979852365041031071…).

If it is, then the Riemann Hypothesis is false. That would be a fun day.

[Taniwha]

5040 is 7! might be a good idea to try a bunch of factorials first ....

[woliveirajr]

I'd love to live in the world where a random Joe would know what ln(ln (n)) is. Or e^y.

[zaarn]

You'll never be sure if you got all the numbers though. You could try 1 trillion numbers and mathmaticians wouldn't be satisfied. Because 1 trillion and 1 could be it.

You'll only know for sure if you prove it false.

[adenadel]

Numberphile has a nice video on 5040 https://www.youtube.com/watch?v=2JM2oImb9Qg

[tzs]

This cites a 2013 paper by Jeffry Lagarias. Lagarias also has a 2001 paper [1] with another elementary equivalent to the Riemann Hypothesis.

Let s(n) be the sum of the positive integers that divide n. For example, s(6) = 1 + 2 + 3 + 6 = 12.

Let H(n) = 1/1 + 1/2 + 1/3 + ... + 1/n.

The Riemann Hypothesis is true if and only if s(n) < H(n) + exp(H(n)) log(H(n)) for all n > 1.

[1] https://arxiv.org/abs/math/0008177

[7373737373]

If (efficiently) searching for a disproof requires fast (prime) factorization, are there connections to other open problems?

[ColinWright]

In this case efficiently searching for an example n does not require fast factorization. We are search for a number n such that σ(n)/(n ln(ln (n))) is large. That mean we want σ(n) large, and that means n will have to have lots and lots of small factors.

So your comment is relevant, but in this case, it doesn't provide new insight.

[lmm]

Oh yes. The Riemann Hypothesis is connected to just about everything; lots of statements have been shown to either follow from it or be equivalent to it.

[FartyMcFarter]

I think you could just generate candidate counter-examples by multiplying factors, rather than the other way around.

[etatoby]

Yes. In fact, if we ever manage to build a quantum computer with a non-trivial amount of qubits, we can expect exponential improvement in the performance of any prime number-related numerical research.

[emmelaich]

5040 == 7!

coincidence?

[impendia]

Not a coincidence.

sigma(n) is largest when n has lots of large factors. Since d is a factor if n/d is, it's easiest to guarantee this by making sure that n has lots of small factors.

Factorials, by construction, are good examples of numbers with many small factors.