Timeline for Analytic continuation of Euler product $\prod_{p} (1 - e^{-2 \pi i p \alpha}p^{-s})^{-1}$

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May 22, 2018 at 4:22	comment	added	Mr. SnowRemover		Okay. As I am interested in the vertical growth of $f(\sigma + it, \alpha)$ for $\sigma > 1/2$, your answer seems to resolve one of my own problems whether it grows at most polynomially when $\alpha$ is rational or not.
May 21, 2018 at 4:47	comment	added	Will Sawin		@Mr.SnowRemover It likely depends on $\alpha$. We can take $\alpha$ a Liouville-type number which is a close approximation to infinitely many rational numbers. By combining this with what I wrote, we should be able to demonstrate that these $\alpha$ have some weird singularity at $s=1$. I suspect for generic $\alpha$ (say, a set of measure $1$) there is analytic continuation to the line $\sigma =1/2$, but don't know if this can be proved.
May 20, 2018 at 23:59	comment	added	Mr. SnowRemover		Thanks. I could learn a lot from your answer. What if $\alpha$ is irrational? With only some basic knowledges on Dirichlet series, I can't even show that there exists an A > 0 such that $f(\sigma + it, \alpha) \ll t^{A} $ as $t \to \infty$ for $\sigma > 1/2$; if we don't know about the behavior of $f(\sigma_{0} + it, \alpha)$ for some $\sigma_{0} < 1$ (its analyticity alone doesn't seem to help), then we can't know anything about $f(\sigma + it, \alpha)$ for $\sigma > \sigma_{0}$ after all, can we?
May 19, 2018 at 6:54	history	answered	Will Sawin	CC BY-SA 4.0