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Just a few days back, Anthropic published a paper that improved the lower bound limit of the non trivial zeros from 41% to 67.2%.
I wrote a simple README for any layman to reproduce this result.
https://github.com/vandan-revanur/claude-67-2-lower-bound/blob/main/how_to_reproduce_67_percent.md
What plane
is on ?
Okay I am starting to understand this a bit better. But I am assuming a prime number would show up in the A variable?
For example
Where 2 would be prime and thus show up on the critical line?
You can think of it as a mapping from a point on a 2d plane to another point on the 2d plane
The inputs are (a,b) and the outputs are (x,y)
Okay I think we did this in my algebra class long ago. I don’t remember it being called the complex plane
Yeah, complex plane is just a generic 2d plane, but with the specific interpretation that each point (x,y) on the plane is basically a complex number: x + yi
And, as to the Riemann zeta function, if we write the inputs as: , then the "critical line" is the line:
The hypothesis says that all the nontrivial solutions to lie on this line.
The values of for which aren't necessarily prime numbers though.
Being an integer on the real number line correct?
I am trying to figure out where the prime numbers show up in the zeta function and how they form this uniform distribution.
I think with a few examples I can finally understand this on a deeper level
I am trying to figure out where the prime numbers show up in the zeta function and how they form this uniform distribution.
all the primes influence each point of the zeta function, and all the zeros of the zeta function influence whether any arbitrary number is prime [although you can obviously do primality testing directly without knowing any zeros, or even how to calculate the zeta function]
there is a relatively old technique [I believe it's due to Euler, who predates Riemann!] for rewriting infinite sums like the zeta function into an infinite product. in the case of the zeta function, that product is over all primes.
getting an intuitive understand for how the behavior of the zeros relates to the distribution of the primes takes much deeper exploration of both the prime counting function, and the zeta function's product expression; and honestly, I haven't understood much of it well enough myself.
is a real number, but not necessarily an integer.
For example, (Here, )
The relationship to prime numbers is more complicated, and not easy to visualize in an intuitive manner. Essentially, the prime counting function (how many primes are there up to some number ), can be expressed as a sum over functions of the non-trivial zeros of the zeta function
Ahhh okay that is why I see the function written like this?
The Reimann zeta function is a complex function which maps a complex number into another complex number, i.e.
The Riemann Hypothesis says that for any input a+bi where ζ(a+bi)=0, either:
How it relates to primes
If the Riemann Hypothesis is true, it says that the primes are distributed relatively uniformly across the real number line.
If the Riemann Hypothesis is false, then it's possible that there are some primes that don't follow a stable distribution.
So, it's related to prime numbers in a deep way, but not necessarily in a very direct way. It doesn't automatically help us find all the primes. It's not like the solution to ζ(a+bi)=0 necessarily is a prime.
Proving or disproving the hypothesis
To prove the Riemann Hypothesis false, you simply have to find an input a+bi where ζ(a+bi)=0 and a=1/2 (excluding all the "trivial" solutions).
To prove the Riemann Hypothesis true, you can't just keep testing inputs because it doesn't prove that there isn't some bigger input which would prove it false. Therefore you have to prove it from previously established general principles, which is much harder and no one has been able to prove.