**Simple question (but not for me):**
Does the Euler product formula diverge for any zero of the Riemann zeta function?


The reason why I ask this is that I heard we should not use the Euler product instead of the Riemann zeta function for Re(s)=<1 because it diverges on the critical strip, but I am not sure of that.


According to my numerical calculation, it seems that it converges for the (known) zeta zeros.


**Additional Question 1:** 
Is it clear that the Euler product for a nontrivial zeta zero is either divergent or convergent?


**Additional Question 2:**
If only one case is possible, which one is the right answer? Divergent or convergent?



Does the Euler product formula diverge for any zero of the Riemann zeta function?
I am in a hurry now but let me tell what I think. I believe that in the critical strip and off the real axis $\prod\_p (1-p^{-s})$ does not converge to any complex number (including zero). Using a similar idea as in my response to your related earlier question, this boils down to the fact that for any nonzero constants $c\_1,\dots,c\_n$ the sum $\sum\_{m=1}^n c\_m \sum\_{p\in P}p^{-ms}$ oscillates wildly as $P\to\infty$. I dont's see this immediately, but I believe what happens is that the oscillation (or divergence) behavior of the inner sum depends heavily on $m$. More precisely, I believe that for $m=1$ you get a much wilder behavior than for the rest $m>1$. So altogether the above double sum inherits the behavior of $m=1$, namely the existence of very large partial sums for infinitely many $P$'s, and this prevents convergence or a tendency to pointing in special directions. I apologize if this is too vague, but certainly more than a comment.


**EDIT 1:** David Speyer showed that in the critical strip $\prod\_p (1-p^{-s})$ does not converge to any *nonzero* complex number, see [here](https://mathoverflow.net/questions/63714/is-the-euler-product-formula-always-divergent-for-0res1). I believe that my approach above can also be made to work and yield more information. Perhaps the Riemann Hypothesis can be of great assistance here as $\sum\_{p\in P}p^{-s}$ is very subtle. Note that for $\mathrm{Re}(s)=1$ and $s\neq 1$ the Euler product does converge to $\zeta(s)$, see Section 3.15 in Titchmarsh: The Theory of the Riemann Zeta-function.


**EDIT 2:** In my response to [this question](https://mathoverflow.net/questions/63714/is-the-euler-product-formula-always-divergent-for-0res1), I outline the proof that, assuming the Riemann Hypothesis, the partial products of $\prod\_p (1-p^{-s})$ get arbitrary close to $0$ and $\infty$, at least for $\frac{1}{2}<\mathrm{Re}(s)<1$. I don't see any fundamental difficulty in extending this to $\mathrm{Re}(s)=\frac{1}{2}$.


