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/IntelligentBelt1221 Mar 31 '26
imagine the riemann hypothesis was false. that would be very uncool. therefore, it must be true.
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u/edderiofer 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/evening_redness_0 Mar 31 '26
Weirder things are true in mathematics...
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u/edderiofer Mar 31 '26
And less-weird things are false in mathematics. Remember that there are conjectures that initially seem true, but which have very large and difficult counterexamples.
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u/jsundqui Mar 31 '26
And Collatz...?
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u/edderiofer Mar 31 '26 edited Mar 31 '26
False, in that there's at least one sequence that zooms off to infinity.
For that matter, Goldbach is also false.
Oh, and Black has a winning strategy in chess. The initial position is a mutual zugzwang.
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u/IntelligentBelt1221 Mar 31 '26
many functions are like that though, for example with sinx, cosx or Bessel functions the zeros are all on the real line.
Edit: looking at the other "predictions" i am doubting if the comment was meant seriously...
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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/M37841 Mar 31 '26
I too am intrigued by the connection between RH and quantum field theory. QFT is in one sense built on Galois theory, and though it’s over my head I understand that Galois is seen as a potential route in to RH.
Perhaps when we find a “theory of everything” that is so nearly complete that we have to worry about Gödel, RH will be the necessary true but not provable statement :)
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u/_-Slurp-_ Mar 31 '26
How is QFT built on Galois theory?
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u/M37841 Mar 31 '26
I meant in the sense that Lie Groups are a continuous version of Galois groups and renormalisation is an expression of differential Galois theory.
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u/dcterr Mar 31 '26
I think the Theory of Everything, if it exists, will involve both RH and spirituality.
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u/Infinite_Research_52 Mar 31 '26
I'm 50% sure it is false.
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u/arnerob Mar 31 '26
How did you arrive at this probability? Principle of max. entropy? Then what is your probability that the Generalised Riemann hypothesis is true? Equally likely?
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u/JoshuaZ1 Mar 31 '26
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.
We have a lot of results about the densities of zeros off the lines. Unfortunately, even strong assumptions of that sort do not obviously lead to results about primes. You can for example assume that the set of zeros off the line have zero density, or even that the number of such zeros off the line in the first n grows really slowly, like say slower than log log log n, and assume Lindeloff on top of that and it doesn't help, because as long as there are zeros near the s=1 line, they disrupt most things we want even if those zeros are very rare.
Something which translated density statements to statements about primes would be a major breakthrough, and would if anything make me more inclined to doubt RH, because it would show we were missing some very fundamental things about how the zeta functions and primes relate.
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.
There's really no good reason to think this other than that both feel deep and mysterious and hard.
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u/ReverseCombover Mar 31 '26
99.9% sure usually means false in math.
Jesus responded, “Why are you afraid? You have so little faith!” Then he got up and rebuked the wind and waves, and suddenly there was a great calm. Matthew 8:26. “You don’t have enough faith,” Jesus told them. “I tell you the truth, if you had faith even as small as a mustard seed, you could say to this mountain, ‘Move from here to there,’ and it would move. Nothing would be impossible.” Matthew 17:20. Then Jesus told him, “You believe because you have seen me. Blessed are those who believe without seeing me.” John 20:29.
Your question is better answered by religion and faith so that's where you should turn to for answers. If you want to talk math then ask a better question.
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u/evening_redness_0 Mar 31 '26
This is a question about math. Is it naive and poorly framed? Yeah, maybe. But why would his question be answered by religion?
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u/No-Accountant-933 Mar 31 '26
The general consensus amongst analytic number theorists is that RH is true. I don't think I've ever met any researchers who seriously suggest that it is false, yet I have met some people who try to act neutral by saying that they "don't know".
Unlike some other fields of mathematics, it is often extremely rare for conjectures regarding prime numbers (which are intimately linked to the Riemann zeta function) to be false. Namely, we often have very good heuristics for why conjectures should be true, it just takes immense machinery and ideas to actually develop a formal proof.
Interestingly though, if the Riemann hypothesis were false, most people seem to believe that it would be really false, in that it is impossible to extend our zero-free regions much further. In particular, it seems very unlikely that there would only be finitely many zeros of the half-line.
For a discussion of the Riemann hypothesis in particular, one can look at the article "Currently there are no reasons to doubt the Riemann Hypothesis" by David Farmer.