Did those have the same sort of theoretical backing as Euler’s argument for the Basel problem?
The point of Jonah’s article is that Euler didn’t just compute a bunch of digits, notice that they matched pi^2/6, and say “done”; he had a theory for why it worked, involving the idea of writing sin(z) as a product of simple factors, and he had substantial (though not conclusive) evidence for that: numerical evidence for other Basel-like sums and a new derivation of Wallis’s product. And he had a bit of confirmation for the general method: he used it to rederive another famous formula for pi. (Though in fact the general method doesn’t work as generally as Euler might possibly have thought it did, as Jonah mentions.)
So far as I know, the only evidence for the Polya and Mertens conjectures was that they seemed to work numerically for small n, and that they seemed like the kind of thing that might be true. Which is plenty good enough reason for conjecturing them, but not at all the same sort of support Jonah’s saying Euler had for his pi^2/6 formula.
The case of Mertens’ conjecture is interesting in that surface level heuristic considerations suggest that it’s not true. In particular, it violates the heuristic that I described here.
Roughly speaking, the function “the difference between the number of natural numbers up to k with an odd number of prime factors and the number of natural numbers up to k with an even number of prime factors” is supposed to be normally distributed with standard deviation ‘square root of k,’ and Merten’s conjecture predicts a truncated normal distribution.
I think a reasonable example is Lamé′s proof of Fermat’s theorem. Experimental evidence confirmed that Fermat’s theorem holds for small numerical examples, and Lamé′s proof shows that if certain rings are unique factorization domains (which is a similar assumption to Euler’s assumption that the product formula for sine holds) then Fermat’s theorem always holds. Unfortunately, the unique factorization assumption sometimes fails.
Granted, the example isn’t perfect because Fermat’s theorem did turn out to be true, just not for the same reasons.
Did those have the same sort of theoretical backing as Euler’s argument for the Basel problem?
The point of Jonah’s article is that Euler didn’t just compute a bunch of digits, notice that they matched pi^2/6, and say “done”; he had a theory for why it worked, involving the idea of writing sin(z) as a product of simple factors, and he had substantial (though not conclusive) evidence for that: numerical evidence for other Basel-like sums and a new derivation of Wallis’s product. And he had a bit of confirmation for the general method: he used it to rederive another famous formula for pi. (Though in fact the general method doesn’t work as generally as Euler might possibly have thought it did, as Jonah mentions.)
So far as I know, the only evidence for the Polya and Mertens conjectures was that they seemed to work numerically for small n, and that they seemed like the kind of thing that might be true. Which is plenty good enough reason for conjecturing them, but not at all the same sort of support Jonah’s saying Euler had for his pi^2/6 formula.
Yes, this is right.
The case of Mertens’ conjecture is interesting in that surface level heuristic considerations suggest that it’s not true. In particular, it violates the heuristic that I described here.
Roughly speaking, the function “the difference between the number of natural numbers up to k with an odd number of prime factors and the number of natural numbers up to k with an even number of prime factors” is supposed to be normally distributed with standard deviation ‘square root of k,’ and Merten’s conjecture predicts a truncated normal distribution.
I agree that Euler’s case was stronger—I was just trying to think of such examples.
I think a reasonable example is Lamé′s proof of Fermat’s theorem. Experimental evidence confirmed that Fermat’s theorem holds for small numerical examples, and Lamé′s proof shows that if certain rings are unique factorization domains (which is a similar assumption to Euler’s assumption that the product formula for sine holds) then Fermat’s theorem always holds. Unfortunately, the unique factorization assumption sometimes fails.
Granted, the example isn’t perfect because Fermat’s theorem did turn out to be true, just not for the same reasons.