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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?

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    $\begingroup$ The keyword here is Deep Riemann Hypothesis. $\endgroup$ Dec 5, 2022 at 10:02

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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.

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    $\begingroup$ I was wondering what the answer to my question would look like in the event that we know the largest real part of the non trivial zeros, let’s say the largest real part is $3/4$. Would the sum then converge for all $s$ with real part greater than $3/4$? And the link you referenced is 40 dollas! $\endgroup$ Dec 2, 2022 at 19:08

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