11
$\begingroup$

Looking at Zagier's Eisenstein Series and the Riemann Zeta Function, we get a proof of the prime number theorem using horocycles. I would really love it if there were a geometric proof like this.

What happens if we use a smaller region of non-vanishing..?
On the fifth page, he show that $\zeta(1 + it) \neq 0$ using properties of the horocycle flow.

If $\zeta(1 + it) = 0$ then $\int_{\Gamma \backslash \mathbb{H}} |\,E\big(z, \frac{1}{2}(1 + it) \big)\,|^2 = 0 $ where $E(z,s) = \sum_{\gamma \in \Gamma_\infty \backslash \Gamma} \mathrm{Im}(\gamma z)^s$

so we can show a zero on the line $\mathrm{Re}(s) = 1$ leads to a contradiction. He wrote a review of Newman's proof , but this result has more geometric flavor.

The next paragraph is on the equidistribution of the horocycle flow. He says the equidistribution of the horocycle flow - with a certain error - can imply the Riemann hypothesis -- that $\zeta(s) \neq 0$ when $\mathrm{Re}(s) \neq \frac{1}{2}$.

$\int_0^1 F(x + iy) \, dx = \frac{1}{\mathrm{Vol}(\Gamma \backslash \mathbb{H})} \int_{\Gamma \backslash \mathbb{H}} F(z) d\mu(z) + O(y^{\frac{1}{2} - \epsilon} ) $

In the modern view, is Zagier's argument rigorous? Is it fair say the equidistribution of the horocycle flow implies the prime number theorem? Certainly he writes one proof on the page, but I think I am seeing two.

I found this blog where Zagier's work is also mentioned with work of Sarnak, Ratner, and Strömbergsson numerous others.

I found Zagier's papers informative but dense and I am not always 100% sure I am reading them correctly. It may be possible that in the process of using short-hand the theorems are false as I have written them!

$\endgroup$

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.