By the way, I've just come across a paper that considers a similar situation: arxiv.org/pdf/0811.1362.pdf . The authors show that $\sum_{n=1}^\infty g(n\alpha)/n^s$ can be analytically continued with (at most) a single pole at $s=1$ if $g$ is 1-periodic, real analytic, and $\alpha$ is Diophantine. Your proof shows that the real analytic hypothesis on $g$ can be weakened at the expense of allowing more poles.

Thanks George. Your rearrangement of $\{k\theta\}^s/k$ is a really nice trick. Some minor typos which don't affect anything: From the second line of the last multi-line equation onwards, $z^{s-j-1}$ should be $z^{s+j-1}$. And by number field, you presumably mean number ring.

That sounds great. I'm looking forward to see your argument.

Regarding the poles on the imaginary axis, the height zeta function in the papers of Tschinkel and Chambert-Loir that Daniel refers to above, do have poles on the line $\Re(s)=1$, so if it is related to $\zeta_\theta$, you might expect something like what I conjecture. It will take me some time to get to grips with their work though.

Thanks for this George. There is a problem though: Your definition of the Hurwitz zeta function excludes $n=0$. So to complete your argument, you'd need to show that \$\sum_{k\neq 0}\{k\theta\}^s k^{-1}\$ (which I can't get to display correctly) extends to a meromorphic function on $\Re(s)<0$. In fact, I think this doesn't converge absolutely for any $s$.

@Daniel: I've thought of using using smooth distance functions, but if $d$ is smooth, $\zeta_d$ only has poles in $\\{1,2,\ldots,n\\}$ (for $d$ a distance function on $\mathbb{R}^n$), whereas if there are non-real poles, as I suspect, you won't be able to find a sequence which converges uniformly on a neighbourhood of these poles. Thinking of this in terms of height zeta functions should be useful though.