Timeline for Which properties of L-functions can be proven assuming they are objects of a symmetric bimonoidal category?
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17 events
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| Nov 21, 2017 at 21:39 | answer | added | reuns | timeline score: 2 | |
| Nov 21, 2017 at 20:48 | comment | added | reuns | No. We are looking at $\mathcal{D} = \{ \sum_{n=1}^\infty a_n n^{-s} = \prod_{j=1}^l L(s,\chi_j)^{e_j}, e_j \in \mathbb{Z}\}$, no sum here. The question is what is $T \mathcal{D}$ and how to put a ring structure on it. The first step is to write a formula for $\log \prod_{j=1}^l L(s,\chi_j)^{e_j}$ and to define $T$ in term of it. | |
| Nov 21, 2017 at 20:30 | comment | added | Sylvain JULIEN | T is a projection : the sum of the images is the image of a sum. | |
| Nov 21, 2017 at 20:22 | comment | added | reuns | And $T f(s) \times Tg(s)$ ? And more important how do you know $T f(s) + Tg(s),T f(s) \times Tg(s)$ are in the image of $T$ ? | |
| Nov 21, 2017 at 20:13 | comment | added | Sylvain JULIEN | Let us continue this discussion in chat. | |
| Nov 21, 2017 at 20:11 | comment | added | Sylvain JULIEN | $ Tf(s)+Tg(s)=\sum_{p}p^{-s}(\sum_{j=1}^{l}e_{j}\chi_{j}(p)+\sum_{k=1}^{m}f_{k}\chi_{k}(p)) $ with $ g(s)=\prod_{k=1}^{m}L(s,\chi_{k})^{f_{k}} $ . | |
| Nov 21, 2017 at 19:55 | comment | added | reuns | First step : if $f(s) = \sum_{n=1}^\infty a_n n^{-s} = \prod_{j=1}^l L(s,\chi_j)^{e_j}, e_j \in \mathbb{Z}$ then $Tf(s)=\sum_p a_p p^{-s} = \sum_p p^{-s} \sum_{j=1}^l e_j\chi_j(p)$. How do you define $Tf(s) + Tg(s)$ and $Tf(s) \times Tg(s)$ to make the image of $T$ a ring ? | |
| Nov 21, 2017 at 19:31 | comment | added | reuns | of course...${}{}$ | |
| Nov 21, 2017 at 19:27 | comment | added | Sylvain JULIEN | @reuns: what exactly do you denote by $\mathcal{P}$? The set of prime numbers? | |
| Nov 21, 2017 at 18:12 | comment | added | reuns | @SylvainJULIEN When considering only products and quotients of Dirichlet L-functions what I wrote is trivial, can you show it ? | |
| Nov 21, 2017 at 10:47 | comment | added | Sylvain JULIEN | @Qiaochu Yuan : I'd be interested in knowing what happens with as many morphisms as possible given this double structure, for different definitions of the tensor product. | |
| Nov 21, 2017 at 5:41 | comment | added | reuns | Of functions $\mathcal{P} \to \mathbb{C}$ quotiented by "equal on all but finitely many primes" (so that it doesn't contain non-primitive Dirichlet L functions) | |
| Nov 21, 2017 at 5:32 | comment | added | reuns | Under the automorphy and Langlands conjectures, isn't the operator $\sum_{n=1}^\infty a_n n^{-s} \mapsto\sum_p a_p p^{-s}$ injective on the products and quotients of normalized L-functions and its image is a ring with the pointwise addition and multiplication of functions $\mathcal{P} \to \mathbb{C}$ | |
| Nov 21, 2017 at 0:23 | comment | added | Qiaochu Yuan | Nothing; with only the given hypotheses $C$ could be a discrete category with only identity morphisms, and then $C$ with this extra structure could be any commutative semiring. | |
| Nov 21, 2017 at 0:23 | comment | added | Noah Snyder | Why would you expect this? What’s the tensor product? What’s Homs from the Riemann zeta to itself? | |
| Nov 21, 2017 at 0:23 | history | edited | Sylvain JULIEN | CC BY-SA 3.0 |
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| Nov 21, 2017 at 0:17 | history | asked | Sylvain JULIEN | CC BY-SA 3.0 |