TITLE: Is the Euler product formula always divergent for 0<Re(s)<1?
QUESTION [21 upvotes]: It is known that the Euler product formula converges for $\Re(s)>1$
(and there it represents the Riemann zeta function).
My question: Is the Euler product always divergent for
$0 < \Re(s) < 1$ ?
I thought that the absolute value of the Euler product formula is positively divergent under the above condition. Is it apparent?

REPLY [3 votes]: The simplest argument is that the domain of convergence of generalized Dirichlet series are always half planes.   Since there is a pole at s=1,  it can only converge for $Re(s) >1$.     
But the question is still very interesting for two reasons.   If you consider instead Dirichlet L-functions based on non-principal characters, then there is no pole at $s=1$ and the above argument does not apply.    In recent work with Franca,  we argued that the Euler product in fact converges for $Re(s)> 1/2$.
The other reason is that for Riemann zeta and other principal characters,  a truncated Euler product can still be made sense of for $Re(s) > 1/2$.
Furthermore,  the Euler product formula is valid in the limit $Im(s) \to \infty$ without truncation in a manner that can be made precise.    This work is on  math.NT