Does the Euler product formula diverge for any zero of the Riemann zeta function?

2011-05-03T09:51:46Z

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?

Answer by GH for Does the Euler product formula diverge for any zero of the Riemann zeta function?

2011-05-03T14:07:45Z

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. 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, 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 $\mathrm{Re}(s)=\frac{1}{2}$.

Does the Euler product formula diverge for any zero of the Riemann zeta function? - MathOverflow

Does the Euler product formula diverge for any zero of the Riemann zeta function?

Answer by GH for Does the Euler product formula diverge for any zero of the Riemann zeta function?