GRH and the Euler product

Question

Let $L(\chi, s)$ be the Dirichlet L-Function of a primitive character $\chi$. I believe, if I’m not mistaken, the convergence of the Euler product of $L(\chi, s)$ in the critical strip is known to be related to GRH. In the event that GRH is false for some character $\chi$, then the Euler product diverges for some zero $\rho = \sigma + it$, with $\sigma \geq 1/2$

My question is, does the sum over primes $p$

$\omega = \sum_{p} \frac{\chi(p)}{p^{\rho}}$

diverge in the sense that the real part of $\omega$ blows up and goes to $-\infty$, or is the real part of the partial sum oscillatory and bounded, or is it oscillatory and unbounded? My hunch has been that it blows up and goes to $-\infty$ with only finitely many sign changes.

And in the event that the zeros of $L(\chi, s)$ cannot get too close to the line $\text{Re}(s) = 1$, that is, there is a vertical strip within the critical strip that acts as a buffer zone between said line and the non-trivial zeros, would the Euler product converge in this buffer zone?

Answer 1

In general, if $F(s)$ and $G(s)$ obeys the relationship

$$ G(s)=\sum_{n\ge1}{F(ns)\over n}, $$

then Möbius inversion gives

$$ F(s)=\sum_{n\ge1}{\mu(n)\over n}G(ns). $$

Using the power series expansion of $\log(1+z)$ near $z=0$, we see that

$$ F(s)=\sum_p{\chi(p)\over p^s},\quad G(s)=\log L(s,\chi). $$

This indicates that we have

$$ P(s,\chi)=\sum_p{\chi(p)\over p^s}=\sum_{n\ge1}{\mu(n)\over n}\log L(ns,\chi). $$

Therefore, the convergence of the Dirichlet series representation for $P(s,\chi)$ would require in-depth knowledge of the distribution of $L(s,\chi)$'s zeros in the critical strip.

Interestingly, current knowledge in the theory of Dirichlet L-functions allows us to conclude that the singularities of $P(s,\chi)$ are dense in the imaginary axis, which indicates that $P(s,\chi)$ does not admit any analytic continuation to the left half plane. See section 9.5 of Titchmarsh's The theory of the Riemann zeta-function and this 1920 paper by Landau and Walfisz for details.