Thread: Infinitely many primes (Riemann Zeta)

1. Infinitely many primes (Riemann Zeta)

Does any one know of a link to the proof that there are infinitely many primes using the Riemann Zeta function or would be willing to produce it?

Thanks.

2. Re: Infinitely many primes (Riemann Zeta)

One of the most 'wonderful' discoveries of the Swiss mathematician Leonhard Euler was the 'product'...

$\displaystyle \sum_{n} \frac{1}{n^{s}} = \prod_{p} \frac{1}{1-\frac{1}{p^{s}}}= \zeta(s)$ (1)

Now if s tends to 1, then $\displaystyle \zeta(s)$ tends to infinity... what's the consequence?...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

3. Re: Infinitely many primes (Riemann Zeta)

Originally Posted by chisigma
One of the most 'wonderful' discoveries of the Swiss mathematician Leonhard Euler was the 'product'...

$\displaystyle \sum_{n} \frac{1}{n^{s}} = \prod_{p} \frac{1}{1-\frac{1}{p^{s}}}= \zeta(s)$ (1)

Now if s tends to 1, then $\displaystyle \zeta(s)$ tends to infinity... what's the consequence?...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$
Well, we need the function to be equal to $\displaystyle \frac{\pi^2}{6}$

And if the Riemann Zeta function goes to infinity, then the sum and product are divergent. So we need s = 2.

Assume there are finitely many primes.

Then

$\displaystyle \zeta(2)\in\mathbb{Q}$

But we know $\displaystyle \zeta(2)\notin\mathbb{Q}$

Therefore, we have reached a contradiction. Correct?

4. Re: Infinitely many primes (Riemann Zeta)

Originally Posted by dwsmith
Well, we need the function to be equal to $\displaystyle \frac{\pi^2}{6}$

And if the Riemann Zeta function goes to infinity, then the sum and product are divergent. So we need s = 2.

Assume there are finitely many primes.

Then

$\displaystyle \zeta(2)\in\mathbb{Q}$

But we know $\displaystyle \zeta(2)\notin\mathbb{Q}$

Therefore, we have reached a contradiction. Correct?
Yes it's correct... but what I had in mind is that the 'infinite sum' $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^{s}}$ when $\displaystyle s=1$ is the 'armonic sum' $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n}$ which diverges to $\displaystyle + \infty$. If the 'discovery' of Leonhard Euler is true, then the product $\displaystyle \prod _{p} \frac{1}{1-\frac{1}{p}}$ diverges to $\displaystyle + \infty$ and that means that the product $\displaystyle \prod_{p} (1-\frac{1}{p})$ diverges to 0. Bur none of the terms $\displaystyle 1-\frac{1}{p}$ is zero, so that necessarly there are infinite factors , i.e. there are infinite primes...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

5. Re: Infinitely many primes (Riemann Zeta)

What is the product of $\displaystyle \zeta(1)$ since the the sum is divergent?

6. Re: Infinitely many primes (Riemann Zeta)

Even if with non precise speaking is $\displaystyle \prod_{p} (1-\frac{1}{p})=\frac{1}{\zeta(1)}=0$...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$