Riemann Hypothesis and Euler product It is conjecture that under certain conditions a L-function satisfies RH.
Among these conditions there is the necessity for the L-function to have an Euler product. (Some L-functions with a functional equation but without Euler product are known to have non trivial zeros with real part between 1/2 and 1).
So the Euler product seems to be an essential ingredient to RH but what are the main properties involved by an Euler product for a L-function ?
(For exemple specific bound known linked to Euler product ? or new relation for the L-function?) To my knowledge the results are very poor.
Note: The equality between L-Function and Euler product holds out of the critical strip but the Euler products directly constraints the non critical zeros to be on the critical line... so I wonder what could be the firts steps of a bridge linking Euler product and RH.
 A: Suppose $L(s)=\sum a_n n^{-s}$ is a Dirichlet series with coefficients satisfying the Ramanujan bound $a_n = O(n^\varepsilon)$ for any $\varepsilon > 0$, and further suppose that $L(s)$ satisfies a suitable functional equation.  Those assumptions are not sufficient for $L(s)$ to have all zeros on the $\frac12$-line, and the question asks for some intuition why the Euler product is the key missing ingredient?
Booker and Thorne consider a set of Dirichlet series as described above, some of which should satisfy RH and some of which should not.  Of those for which RH is expected to fail, they show that it fails in the most spectacular way possible:  the function has infinitely many zeros in the region $\sigma >1$.  (As usual, $s=\sigma + i t$ with $\sigma$ and $t$ real.)  And the functions which should satisfy RH have an Euler product, and so no zeros in $\sigma >1$.
So, it is possible that the assumption of an Euler product can be replaced by an assumption that the function has no zeros in $\sigma \ge 1$.  In other words, once you push the zeros into the critical strip, they must go all the way to the critical line.
Since the (absolute convergence of the) Euler product implies nonvanishing for $\sigma > 1$, it is possible that the Euler product is just a proxy for that nonvanishing.
A: The conditions to impose to the Euler product of an L-function in order for a generalized Riemann hypothesis to hold are well understood after Selberg's 1992 paper.
In particular, given $L(s)=\prod_p L_p(s)$, it should be,
$$\log L_p(s)=\sum_{n=1}^\infty \frac{b_{p^r}}{p^{rs}}$$
and
$$b_{p^r}=O(p^{n\theta})$$
with $\theta<1/2$.
These conditions are neccesary (Dirichlet $\eta$ function is the standard counterexample), but together with three other assumptions (analyticity, a Ramanujan-Petterson type bound on the coefficients, and an appropiate functional equation) are also expected to be sufficient.
