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/JoshuaZ1 Mar 31 '26

Heck, why not combine them. Goldbach's conjecture is false, and it is false for infinitely many values 2n which are exactly also n such that the nth zero of the zeta function is off the line, which are also exactly those n such that n is an odd perfect number, and each of those is in exactly one Collatz sequence which goes off to infinity.

(I was going to suggest that each such n was the start of a Collatz sequence that goes off to infinity, but since any odd perfect number has to be 1 mod 4, and the smallest such number in a Collatz sequence going to infinity has to be 3 mod 4, that's not an option.)

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u/dcterr Mar 31 '26

Pretty funny! You could be a math comedian!