This came up on my feed a few days ago and I’m still trying to understand the theory and importantly the proof needed to claim the $1M.
If i understand this correctly this Riemann guy used complex numbers and put the inverse ratio into a zeta function to find where the zeros are and every time a zero is found relates to finding a prime number.
(Sorry for the crappy explanation but I am trying to understand it in my own words to help my brain get a better understanding)
Everyone in mathematics is excited because we have a function to predict prime numbers correct? But what proof is needed that this method actually does work? I saw other YouTube videos say computers have calculated like a trillion prime numbers but still isn’t enough proof to win the prize? Why the heck not??
If anyone takes time to explain this ill zap them lots SATS!
I think I'll build a Riemann zeta function visualizer. Coming soon!
I am wondering can you do this with pencil and paper?
you'll need either at least two colors, or some skill with grayscale sketching [and possibly multiple pencil guages]
riemann zeta is a complex function, so any point on the 2d plane of your sketch, needs to get multiple dimensions of information drawn... this is in total four dimensions, and can't be captured accurately without encoding an additional dimension somehow. one typical method that is quite intuitive once you're familiar with it is hue [red/green/blue] to represent complex phase [e.g. red is positive, cyan is negative, green and blue are the imaginary cube roots of 1], and then brightness or saturation to represent magnitude.
geek out a bit on youtube about complex functions [e.g. gamma is another common one] and you'll see tons of examples. they can be drawn manually, it might even boost your intuition much more effectively than passively watching videos... it's just a lot of delicate work and computers do it better.
the work done by 3blue1brown on visualizing Riemann zeta function on the complex plane is beautiful. Take a look at the Manim library.
https://github.com/3b1b/manim
I am assuming this can’t be done on an X-Y plane?
it can be rendered on a x-yi plane
linking riemann to black holes triggers all my clickbait filters; however, there are some few cases of what I call "practical numerology", where something from pure mathematics turns out to have implications on the shape of the most-practical bits of theoretical physics... the most notable example that springs to mind is the "monstrous moonshine conjecture".
I like this kind of link because it gives me hope that abstract wankery along the lines of "you can't prove the universe is not a simulation!" can get brushed aside by the much more useful "if the universe actually is a simulation, here's what can be said about efficiency optimisations taken by the computational substrate".
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 primesHow 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 hypothesisProving 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.
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: a+bi, then the "critical line" is the line:
The hypothesis says that all the nontrivial solutions to ζ(a+bi)=0 lie on this line.
The values of b for which ζ(21+bi)=0 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
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.
b is a real number, but not necessarily an integer.
For example, ζ(21+14.134i)=0 (Here, b=14.134...)
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 x), 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?
It doesn’t matter how many they find, a formal proof has to show conclusively that it’s logically impossible to generate a non-prime with this function.
There are lots of conjectures that hold over every number that’s been tried but that remain unproven.
But how do we show that? and let’s say we do generate a non-prime that will mean Riemann’s hypothesis is wrong?
To the latter question, yes, that would mean the hypothesis is wrong (assuming it's described accurately in the post).
As to how to logically prove a theorem, there are a bunch of possible approaches. The two easiest to describe are direct proof and proof by contradiction:
Direct proofs are often more difficult than proofs by contradiction, at least in my experience.
unfortunately your oversimplification is misleading to the point of being incorrect.
the riemann zeta function isn't about any single prime, or whether any single arbitrary number is or is not prime. it's about the distribution of all the primes, considered as a single mathematical spectrum.
What do you mean exactly when you say this
Alright so primes are weird. 2, 3, 5, 7, 11... they just show up randomly with no obvious pattern and mathematicians have been trying to predict them forever.
This guy Riemann in the 1800s built a mathematical machine that seems to track where primes hide. When you run the machine, every so often the output hits exactly zero. Those moments are special. And Riemann noticed every single one of those zero moments landed on the exact same imaginary line, dead center, every time. He couldn't prove it always happens. He just noticed it and reckoned it was always true. That's the whole hypothesis.
Now here's why checking a trillion examples doesn't win the million dollar prize. In normal life a trillion confirmations and you'd bet your house on it. But math deals in infinity and a trillion out of infinity is basically nothing. Math has also been burned before by patterns that held for millions of cases and then randomly broke.
Mathematicians don't want to know that it seems true. They want to know WHY it's impossible for it to ever be false. Like you don't check every triangle ever drawn to prove triangles can't have four sides. You just show that four sides breaks the whole definition of a triangle. Someone needs to write a pure logic argument showing a zero landing off that line is a structural impossibility, not just something nobody's seen yet.
Most mathematicians think that argument is going to need ideas that haven't been invented yet. A 160 year old problem still waiting for math that doesn't exist.
And here's the wildest part. Equations that describe how black holes scramble information produce patterns that look identical to Riemann's zeros. Nobody fully understands why. But it suggests the proof might not come from a mathematician with a notebook. It might come from a physicist working backwards from a black hole.
The secret pattern in prime numbers, unlocked by a black hole. Not a bad story.