I Is Riemann's Zeta at 2 Related to Pi through Prime Numbers?

[Jarvis323]

Another fascinating relationship is to do with points generated by repeated multiplication and modulus plotted around a circle.



These are sequences generated based on number theoretic phenomena that is related to primality and divisibility. And they form the same patterns you get if you roll circles around other circles. Which also happen to be the shapes of magnetic fields in microphones, and a basis for forming Fibonacci spirals, and the shape of the Mandelbrot fractal.

 

[Janosh89]

In #11 PeroK mentions finite series,

as someone who has rudimentary maths & is recovering from a brain injury ,
I'm assuming these series converge to some value , π in the instance the OP
alluded to.

Infinite Series are treated in some other way [ eg 12x +1 which contains all the
primes of the form 6x +1 ] ?

 

[PeroK]

In #11 PeroK mentions finite series,

I mentioned finite sequences. A sequence is a list of numbers; a series is a sum. That's the formal mathematical usage of those terms.

I'm assuming these series converge to some value , π in the instance the OP
alluded to.

All finite series have a finite sum. It's only infinite series where we need to talk about convergence or divergence.

Infinite Series are treated in some other way [ eg 12x +1 which contains all the
primes of the form 6x +1 ] ?

I don't know what you mean here.

 

[Janosh89]

Please take down my post if it does not contribute to the thread

 

[martinbn]

I might repeat what is already said but one has of course

$$\prod_{p\; prime}\frac1{1-p^{-2}} = \frac{\pi^2}{6}$$

which also proves that there are infinitely many prime numbers. Otherwise the left side is a rational number.

 

[dextercioby]

I might repeat what is already said but one has of course

$$\prod_{p\; prime}\frac1{1-p^{-2}} = \frac{\pi^2}{6}$$

which also proves that there are infinitely many prime numbers. Otherwise the left side is a rational number.

Can you point to a proof of that?

 

[martinbn]

Can you point to a proof of that?

That is Riemann's zeta at 2. It is in many places. Here is one of the first that come up.

https://www.math.cmu.edu/~bwsulliv/MathGradTalkZeta2.pdf

 

[dromarand]

$\prod_{p \ prime}{} \frac{p^{2}+1}{p^{2}-1} = \frac{5}{2}$
$\prod_{p \ prime}{} \frac{p^{4}+1}{p^{4}-1} = \frac{7}{6}$

 

[WWGD]

Can you point to a proof of that?

You can also show this using Fourier Series for $x^2$ in $[-\pi, \pi ]$
https://math.stackexchange.com/ques...m-n-1-infty-1-n2-using-fourier-series#2706955