Timeline for Meromorphic continuation of a Dirichlet series associated to an irrational number
Current License: CC BY-SA 3.0
14 events
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| Apr 8, 2013 at 19:31 | comment | added | George Lowther | @Pieter: Thanks, I fixed those typos. Also extended the argument to arbitrary quadratic irrationals. | |
| Apr 8, 2013 at 19:30 | history | edited | George Lowther | CC BY-SA 3.0 |
Extended proof. Fixed typos.
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| Apr 8, 2013 at 2:25 | comment | added | Pieter | 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. | |
| Apr 8, 2013 at 2:10 | comment | added | Pieter |
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.
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| Apr 8, 2013 at 2:06 | vote | accept | Pieter | ||
| Apr 5, 2013 at 23:38 | history | edited | George Lowther | CC BY-SA 3.0 |
deleted 105 characters in body; deleted 1 characters in body
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| Apr 5, 2013 at 23:30 | history | edited | George Lowther | CC BY-SA 3.0 |
Updated proof
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| Apr 5, 2013 at 21:03 | comment | added | Pieter | That sounds great. I'm looking forward to see your argument. | |
| Apr 4, 2013 at 14:53 | comment | added | George Lowther | @Pieter: I think I know how to handle this term now, and you get poles on the infinite set of lines $\lbrace -2n+i\mathbb{R}\colon n\in\mathbb{N}\rbrace$, not just the imaginary axis. The argument is a bit rough, but I'll update my answer if I can log on from home tonight. | |
| Apr 3, 2013 at 13:34 | comment | added | George Lowther | You're right, I missed out the n=0 term, which gives something like $\sum\_{k\not=0}\lbrace k\theta\rbrace/k$. This converges on $\Re(s) > 1$ and would need to be extended. It seems entirely plausible that it would extend, with poles on the imaginary axis, but I'm not sure how you would prove that. | |
| Apr 3, 2013 at 4:05 | comment | added | Pieter | 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. | |
| Apr 3, 2013 at 4:03 | comment | added | Pieter |
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 $s$.
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| Apr 3, 2013 at 2:25 | history | edited | George Lowther | CC BY-SA 3.0 |
added 31 characters in body
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| Apr 3, 2013 at 0:14 | history | answered | George Lowther | CC BY-SA 3.0 |