# My Works on Abelian Zetas

## Since 2009

 $\huge{\color{Brown}{\mathbb{\ Abelian\ Zeta\ Functions\ For\ (G,P)/Q}}}$  $\bigodot\ \Large{G:\ \mathrm{a\ reductive\ group\ with\ P\ its\ maximal\ parabolic\ subgroup,\ all\ defined\ over}\ \mathbb Q }$  $\bigodot\ \Large{\Phi^\pm:\ \mathrm{the\ associated\ system\ of\ positive,\ resp. negative, roots} }$  $\bigodot\ \Large{\Delta:\ \mathrm{the\ set\ of\ simple\ roots} }$  $\bigodot\ \Large{W:\ \mathrm{the\ Weyl\ group} }$  $\bigodot\ \Large{\rho\,:=\,\frac{1}{2}\sum_{\alpha\in\Phi^+}\alpha\ \mathrm{the\ Weyl\ vector} }$  $\bigodot\ \Large{\{\beta_1,\beta_2,\dots,\beta_{|G|-1}\}:=\Delta-\{\alpha_P\} \ \mathrm{with}\ \alpha_P\ \mathrm{the\ simple\ root\ corresponding\ to}\ P }$  $\huge{\heartsuit\ \color{Red}{\mathrm{Weng\ Zeta\ Functions\ for\ (G,P)}/\mathbb Q:}}$ \boxed{\begin{align*}&\Large{\widehat\zeta^{G/P}_{\mathbb Q}(s):=}\\ &:=\Large{\mathrm{Norm}}\,\Bigg({\Large\mathrm{Res}}_{\Large{\langle \lambda-\rho,\beta_1^\vee\rangle=0,\,\langle \lambda-\rho,\beta_2^\vee\rangle=0,\, \dots,\, \langle \lambda-\rho,\beta_{|G|-1}^\vee\rangle=0}}\,\Big( {\Large\sum}_{w\in W}\frac{\Large{e^{-\langle \lambda,T\rangle}}}{{\huge\prod}_{\alpha\in\Delta}\langle \lambda-\rho,\alpha^\vee\rangle}\cdot{\Large\prod}_{\alpha\in \Phi^+, w\alpha\in\Phi^-} \frac{\widehat\zeta\big(\langle\lambda,\alpha^\vee\rangle\big)}{\widehat\zeta\big(\langle\lambda,\alpha^\vee\rangle+1\big)}\Big)\Bigg) \end{align*}}   $\huge{\spadesuit\ \color{Blue}{\mathrm{Functional\ Equation:}}\qquad \exists\qquad {constant}\quad c_{G/P},\quad \mathrm{s.t.}}$ $\boxed{\huge{ \widehat\zeta^{G/P}_{\mathbb Q}(c_{P/Q}-s)=\widehat\zeta^{G/P}_{\mathbb Q}(s)}}$   $\huge{\spadesuit\ \color{Blue}{\mathrm{Riemann\ Hypothesis:}}}$ $\boxed{\huge{\widehat\zeta^{G/P}_{\mathbb Q}(s)=0\qquad\Rightarrow\qquad\mathrm{Re}(s)=\frac{c_{G/P}}{2} }}$