Timeline for $L$-functions for quadratic orders and Siegel's solution of the class number problem
Current License: CC BY-SA 4.0
3 events
| when toggle format | what | by | license | comment | |
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| Nov 9, 2018 at 16:58 | comment | added | Shimrod | @reuns Could you please elaborate on your comment? | |
| Nov 9, 2018 at 0:11 | comment | added | reuns | A representative of $\mathfrak{c}$ is a lattice $ u \mathbb{Z}+v \mathbb{Z}, u/v = a+ib$ and you can look at $h(x) = \sum_{n \in \mathbb{Z}^2} \exp(-\pi x \|\begin{pmatrix} 1 & a \\ 0 & b \end{pmatrix} n\|^2)$. The Poisson summation formula tells you $h(1/x) = |b|^{-1/2} x^{-1} \sum_{n \in \mathbb{Z}^2} \exp(-\pi x \| \begin{pmatrix} 1 & a \\ 0 & b \end{pmatrix}^{-\top} n\|^2)$ (which is related to the sum for a representative of $\mathfrak{c}^{-1}$). From this you know the asymptotic of $h$ as $x \to 0$ and the pole of its Dirichlet series, that you can relate to $\log \eta(u/v)$ | |
| Nov 8, 2018 at 19:39 | history | asked | Shimrod | CC BY-SA 4.0 |