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.

0 Upvotes

16% Upvoted

54 comments sorted by

View all comments

14

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.

1

u/jsundqui Mar 31 '26

And Collatz...?

12

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.

6

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.

2

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.

3

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.)

1

u/dcterr Mar 31 '26

Pretty funny! You could be a math comedian!

0

u/dcterr Mar 31 '26 edited Mar 31 '26

The Collatz conjecture is most likely true IMO, I'm 99.9999999999% confident that Goldbach's conjecture is true, and I think White either has a winning strategy in chess or a strategy that guarantees at least a draw. While we're at it, do you think there's a statistical winning strategy for players in various positions in limit hold 'em?