Timeline for Does this product have analytic continuation?
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Sep 4, 2017 at 18:33 | comment | added | reuns | @GHfromMO Hi, I'm trying to understand the theory behind the convergence of $\lim_{k \to \infty} \sum_{n=1}^\infty a_n n^{-s} 1_{gcd(n,N_k)=1}$ for arbitrary Dirichlet series with functional equation, with $ N_k = \prod_{p \le k}$. From that I obtain heuristics suggesting a proof of the RH would imply the analyticity of $\sum_p p^{-s} e^{2i \pi \xi p}$ for $\Re(s) > 1/2, \xi \in (0,\pi)$ (and hence GRH). Do you know this subject, and what it could be useful for compared to analyticity of $\sum_p p^{-s} \chi(p)$ ? | |
| Jul 19, 2017 at 7:34 | vote | accept | CommunityBot | ||
| Jul 18, 2017 at 12:46 | comment | added | GH from MO | I agree though that $F(s)^{N!}$ is meromorphic for $\Re(s)>1+\frac{1}{N+1}$. This is a nice observation and extends what I said. | |
| Jul 18, 2017 at 12:43 | comment | added | GH from MO | I think a branch point already prevents analytic continuation as a single valued function to a half-plane minus a discrete set of points of isolated singularities. So $P(s)$ already cannot be continued beyond the $\Re(s)=1$ line due to the logarithmic singularity of $\log\zeta(s)$ at $s=1$. | |
| Jul 18, 2017 at 12:31 | history | edited | reuns | CC BY-SA 3.0 |
added 1 character in body
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| Jul 18, 2017 at 11:56 | history | answered | reuns | CC BY-SA 3.0 |