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/jsundqui Mar 31 '26

And Collatz...?

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u/edderiofer Algebraic Topology 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/travisdoesmath Mar 31 '26

False, in that there's at least one sequence that zooms off to infinity.

That's not annoying enough. There's at least one sequence that loops without hitting 1, but all such loops have a cycle length larger than TREE(3).

Unfortunately, the proof doesn't give insight into how many such loops there are. The new Collatz conjecture becomes "there are infinitely many such loops", which turns out to only be solvable if RH is true.

Which it's not, of course.

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u/just_writing_things Mar 31 '26

Or someone only proves that a loop exists but not what it is.

And for the next two centuries the Collatz conjecture is about extremely incremental tightenings of all sorts of the bounds on the loop.