If the Riemann hypothesis is false, how many counterexamples does it have?
3
Ṁ150Ṁ2282999
9%
Four
9%
Eight
13%
Twelve
26%
More than twelve, but finitely many
37%
Countably infinitely many
5%
Uncountably many
If s is a counter-example (i.e. ζ(s) = 0 but −s/2 ∉ ℕ and ℜ(s) ≠ 1/2), then so are s* (the complex conjugate), 1−s, and 1−s*, so the answer must be a multiple of four.
Resolves N/A if the Riemann hypothesis is true.
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The Riemann zeta function is analytic everywhere except at s=1. Therefore, as you learn in your very first complex analysis class, its zeros must be isolated, ie around any zero there is a disk containing no other zeros. And as you learn in your first real analysis or topology class, this means there can only be countably many zeros.
So, the "uncountably many" option is mathematically impossible.
@pietrokc well my first complex analysis class was one and a half decade before I created this market
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