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Let $F$ be a quadratic number field with class number $h_F = 1$. Let $\zeta_F$ be the Dedekind zeta function, we have $$ \frac{\zeta_F ' (1+it)}{\zeta_F (1+it)} \ll \frac{\log t}{\log\log t} .$$ (I do …

Let $F$ be an imaginary quadratic number field. Let $G = \mathrm{PGL}_2(\mathbb{C}) $ and $\varGamma = \mathrm{PSL}_2(\mathcal{O}_F)$. We have the Iwasawa decomposition $G = NAK$ where $K = \mathrm{PS …

Let $F = \mathbb{Q}(\sqrt{-d})$ with class number $h_F = 1$, and $\Gamma = \mathrm{PSL}_2(\mathfrak{O}_F)$. Let $f$ be a Maass cusp form in the $L^2$-cuspidal spectrum of the Laplace operator $\v …

Recently, I came across the Langlands correspondence theorem, there is the following line: $$L(s,\pi(\sigma) \times \pi(\tau)) = L(s,\sigma \otimes \tau), $$ where $\sigma$ and $\tau$ are representat …

I am currently reading Garrett's book "Holomorphic Hilbert Modular Forms". But I meet trouble at the starting line. Let $F = \mathbb{Q}(\sqrt D)$ be a real quadratic field, $u= a + b\sqrt{D}\in F$, an …