The $\zeta$ function maybe written as Euler Product: $ \zeta(s)=\prod_{p} \frac{1}{1-p^{-s}}=\prod_p e_p(s). $ Now let's substitute $s$ with $\rho_k$, the $k$th root of $\zeta$, and have a look at the individual factors. There are 2 options: 1.$|e_p(\rho_k)|<1$ or 2. $|e_p(\rho_k)|\ge1$ (e.g. $|e_{23}(\rho_1)|\approx 1.2404$, for more values, see here) and obviously the values less $1$ must be infintely many more than the others, otherwise it would not converge to $0$. So we can write it as: $ \zeta(\rho_k)=\prod_{\color{blue}{||<1}} e_p(\rho_k) \times \prod_{\color{red}{||\geq 1}} e_p(\rho_k), $
So my questions are:
- Do these values $e_p(\rho_k)$ have a special meaning or a straight forward interpretation?
- For a given $\rho_k$, how do these values distribute? Plots for $\rho_1$ and $\rho_2$ (not shown) show a spiral around $1+0i$:
($x$ is real axis, $y$ the imaginary, the line indicates $||=1$)
- obsolete Are there finitely or infinitely many primes $p$, where $|e_p(\rho_k)|\ge1$? Infinitely.
- How does that distribution behave, if $\Re(s)\neq \frac{1}{2}$? Some values of $e_p(\varepsilon + \rho_k)$ should move outwards, such that $\prod_{||<1} e_p(\rho_k)$ doesn't become $0$ anylonger, if Riemann's Hypothesis is true. $s=1/8+14.134725i$ is shown in the plot:
Thanks for your help/comments/plots/answers...