Timeline for Meromorphic continuation of a Dirichlet series associated to an irrational number
Current License: CC BY-SA 3.0
9 events
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| Apr 12, 2013 at 2:22 | history | edited | Pieter | CC BY-SA 3.0 |
Reference to solution of problem by Hecke
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| Apr 8, 2013 at 2:06 | vote | accept | Pieter | ||
| Apr 3, 2013 at 0:14 | answer | added | George Lowther | timeline score: 8 | |
| Apr 2, 2013 at 7:45 | comment | added | Daniel Loughran | @Pieter: Tschinkel has done lots of work on height zeta functions. In particular I think that his recent work with Chambert-Loir on integral points of bounded height on toric varieties applies to your setting (see his webpage). The analytic behaviour of these zeta functions is closely related to corresponding number of rational/integral points of bounded height. In their paper they use smooth norms like you, but I seem to remember some trick which allows you to pass to non-smooth norms as I mention above, however the precise details of this trick currently elude me... | |
| Apr 1, 2013 at 19:58 | comment | added | Pieter | @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. | |
| Apr 1, 2013 at 18:18 | comment | added | George Lowther | We can clearly write $\zeta_\theta(s)=\theta\zeta(s-1)+f(s)$ where $f$ is analytic on $\Re(s) > 1$. Furthermore, by equidistribution, $f(s)=-1/(2(s-1))+o((s-1)^{-1})$ as $s\to1$. Then, I think, you can show that $f(s)+1/(2(s-1))$ is unbounded as $s\to1$ for all $\theta$ outside of a meagre set. That implies that it is not possible to extend to a meromorphic function on $\Re(s) > \sigma_0$ for any $\sigma_0 < 1$, for all $\theta$ outside of a meagre set. However, this says nothing about your tentative conjecture in the last sentence, as the algebraic numbers are meagre. | |
| Apr 1, 2013 at 8:55 | comment | added | Daniel Loughran | Also, do you know about Height zeta functions and Epstein zeta functions? The zeta functions $\zeta_d$ which you are studying are very closely related to these. For example your zeta function with $d(x,y)=\mathrm{max}\{|y|,|x|/\theta\}$ is a height zeta function for integral points in the affine plane. Here meromorphic continuation is known in some region past $s=2$ (though I am not sure if meromorphic continuation is know to the whole complex plane). | |
| Apr 1, 2013 at 8:03 | comment | added | Daniel Loughran | Here is one idea for studying $\zeta_d$ with $d$ not smooth: Find a sequence of smooth norms $(d_t)$ with $t \in \mathbb{R}$ which converge to the norm $d$ you are interested in. If you have enough control over the convergence you might be able to deduce the meromorphic continuation of $\zeta_d$ from the meromorphic continuation of each $\zeta_{d_t}$. | |
| Apr 1, 2013 at 3:54 | history | asked | Pieter | CC BY-SA 3.0 |