Timeline for Functional equation link two Dirichlet series
Current License: CC BY-SA 3.0
14 events
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| Sep 12, 2016 at 20:03 | comment | added | reuns | you mean $M(s) \approx \log R_f(s)$, and what I wrote should give $\sum_p |a(tp)|^2 p^{-s} \approx \log(\sum_{n=1}^\infty |a(tn)|^2 n^{-s})$ | |
| Sep 12, 2016 at 19:58 | comment | added | Med | @user1952009, That's not what I am looking for :-) I am interested to a functional equation, look like $M(s)=\Delta(s)R_f(s)$ | |
| Sep 12, 2016 at 19:42 | comment | added | reuns | reading Shimura, on modular forms of half integral weight , theorem 1.9 p.13 it says $\sum_{n=1}^\infty a(tn) n^{-s}$ has an Euler product whenever $t $ is square-free and $> 1$ | |
| Sep 12, 2016 at 18:42 | history | edited | Med | CC BY-SA 3.0 |
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| Sep 12, 2016 at 18:34 | history | edited | Med | CC BY-SA 3.0 |
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| Aug 13, 2016 at 17:51 | comment | added | paul garrett | Ah, indeed, I did overlook the half-integral aspect... but, still, your $R_f$ could be obtained by integrating $|f|^2$ against suitable $E_s$ as usual. And I'd still be very surprised if just taking prime-index components didn't have a natural boundary. | |
| Aug 13, 2016 at 17:42 | comment | added | Med | @user1952009, We don't really have an Euler product in the half-integral weight case !!! | |
| Aug 13, 2016 at 17:12 | comment | added | Med | @paulgarrett , @ user1952009, I think that you are overlooked that f is a modular form of half-integral weight ....!!! | |
| Aug 13, 2016 at 2:39 | comment | added | reuns | Hecke eigenform means $R_f(s)$ has an Euler product $R_f(s)= \prod_p 1+\sum_{k=1}^\infty a(p^k)^2 p^{-sk}$, so that $\log R_f(s) = \mathcal{O}(1) + \sum_p a(p)^2 p^{-s}+ a(p^2)^2 p^{-2s}$ for $Re(s) > k+1/2+\epsilon$ (or something like that) | |
| Aug 12, 2016 at 22:47 | comment | added | Med | Dear @paulgarrett, Could you please explain to me what you mean by "The first displayed expression is a fragment of the logarithm of the second," | |
| Aug 12, 2016 at 22:03 | comment | added | Med | @paulgarrett, Thank you very much for your comment. | |
| Aug 12, 2016 at 21:50 | comment | added | paul garrett | The second is approximately the Rankin-Selberg convolution of the standard $L$-function attached to $f$ with itself. It has a functional equation. The first displayed expression is a fragment of the logarithm of the second, and I'd wager that it has a natural boundary... | |
| Aug 12, 2016 at 21:16 | history | edited | Med | CC BY-SA 3.0 |
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| Aug 12, 2016 at 21:07 | history | asked | Med | CC BY-SA 3.0 |