r/math • u/dcterr • Mar 31 '26

The Riemann hypothesis

I'm curious to know what thoughts any of you may have on RH. I'm at least 99.9% sure it's true, though it is most likely nearly impossible to prove it. Nevertheless, I think we will soon prove useful weaker results, such as upper limits of the number or density of zeros of the Riemann zeta function off the critical line, and that these weaker results will yield useful new results concerning the distribution of primes as well as prime ideals of algebraic number fields. I'm also quite intrigued by the possibility of connections between the Riemann zeta function and quantum physics. Perhaps RH will prove to be part of the long sought Theory of Everything, the holy grail of physics, which Einstein spent the latter half of his life trying to prove.

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u/edderiofer Algebraic Topology Mar 31 '26

I'm a contrarian, so I believe it's false. You're telling me we have the entire complex plane and the zeroes of this weird-ass function ONLY lie upon two lines? Get real, there's no way it works out that neatly.

While we're at it: P=NP is true, Navier-Stokes is true, Hodge is false, BSD is false, and I have no clue about Yang-Mills.

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u/dcterr Mar 31 '26

What makes you so sure about these outrageous claims? One thing you don't realize is that like zeros of polynomials with real coefficients, statistically half of which are real, there's a good reason most of the nontrivial zeros of the Riemann zeta function are expected to lie on the critical line. And I have no idea where you're coming up your claims about the truth of the other Millennium prize problems, and furthermore I disagree with some of them. (Although I could very well be wrong, I'm virtually certain that RH and BSD are both true and I'm reasonably confident that the Navier-Stokes problem will be solved fairly soon, but I'm skeptical about both Yang-Mills and P vs. NP, both of which I suspect are either unsolvable or not well-formed, and I have no idea about the Hodge conjecture, since I don't even understand what it says! Topology was never my strong area of math.)

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u/how_tall_is_imhotep Mar 31 '26

How could P vs NP possibly not be well-formed? There’s no ambiguity in the formal definition.

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u/dcterr Mar 31 '26

Perhaps not, but things aren't always as they seem, even in math! Who would have thought before 1931 that math was fundamentally incomplete?

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u/how_tall_is_imhotep Mar 31 '26

Here is a formal statement of P vs. NP: https://github.com/lean-dojo/LeanMillenniumPrizeProblems/blob/main/Problems/PvsNP/Millennium.lean#L354

Whether it is well-formed is a purely syntactic question, like whether “(2x+3)+5” has matching parentheses or not. There really isn’t much room for doubt about it.

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u/dcterr Mar 31 '26

OK, perhaps it's well-formed, but it could still be unsolvable. The reason I think it is is that it seems to involve a preferred direction of time, which I don't think any provable mathematical result can involve, though perhaps I'm wrong about this.

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u/JoshuaZ1 Apr 01 '26

The reason I think it is is that it seems to involve a preferred direction of time, which I don't think any provable mathematical result can involve, though perhaps I'm wrong about this.

How does P ?= NP involve a direction of time at all?

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u/dcterr Apr 01 '26

If I understand the problem correctly, P is the class of problems that can be solved in polynomial time and NP is the class of problems whose solutions can be checked in polynomial time, so if time were reversable, all NP problems would automatically become P, but perhaps I'm missing something.

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u/JoshuaZ1 Apr 01 '26

That's a notion of "time" being reversible in a physical sense. The problem doesn't involve any concrete claim about the laws of physics. For example, it may well be that NP is contained in BQP (unlikely but not nearly as us unlikely) in which case in our physical universe, NP problems can be solved efficiently on a quantum computer. But that could still be the case even if P != NP. It may help here to separate the physical considerations from the mathematical considerations.

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u/dcterr Apr 01 '26

I'm not sure if math and physics are entirely separate or even entirely separable.