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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?
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)