# My Works on Non-Abelian Zetas

## Since 1999

 $\huge{\color{Brown}{\mathbb{\ High\ Rank\ Zeta\ Functions}}}$  $\bigodot\ \Large{F:\ \mathrm{algebraic\ number\ field\ with}\ \mathcal O_F\ \mathrm{the\ integer\ rings\ and}\ \Delta_F\ \mathrm{the\ absolute\ value\ of\ discriminant}}$  $\bigodot\ \Large{\mathcal M_{F,n}\ (\mathrm{resp}.\ \mathcal M_{F,n}[\geq 1],\ \mathcal M_{F,n}[1]):\ \mathrm{moduli\ space\ of\ s.stable}\ \mathcal O_F\,\mathrm{lattices\ of\ rank}\ n,\ \mathrm{(resp.\ of\ volume}\ \geq 1,\ \mathrm{of\ volume}\ 1)}$  $\bigodot\ \Large{h^0(F,\Lambda):\ \mathrm{the\ 0-th\ geo-arithmetic\ cohomology\ of\ the\ lattice}\ \Lambda }$  $\bigodot\ \Large{\mathrm{deg}(\Lambda):\ \mathrm{the\ Arakelov\ degree\ of\ the\ lattice}\ \Lambda }$   $\huge{\heartsuit\ \color{Red}{\mathrm{Non-Abelian\ Zeta\ Functions:}}}$  $\boxed{\LARGE{\displaystyle{\widehat\zeta_{F,n}(s):=\big(\Delta_F^s\big)^{\frac{n}{2}}\int_{\Lambda\in\mathcal M_{F,n}}\Big(e^{h^0(F,\Lambda)}-1\Big)\cdot\big(e^{-s}\big)^{\mathrm{deg}(\Lambda)}\,d\mu(\Lambda),\qquad\mathrm{Re}(s)>1}}}$   $\huge{\clubsuit\ \color{Blue}{\mathrm{Functional\ Equation\ \&\ Singularities:}}}$  \boxed{\LARGE{\begin{aligned}\widehat\zeta_{F,r}(s)= &\int_{\Lambda\in\mathcal M_{F,n}[\geq 1]}\Big(e^{h^0(F,\Lambda)}-1\Big)\cdot\mathrm{Vol}(\Lambda)^s \, d\mu(\Lambda)\\ &+\int_{\Lambda\in\mathcal M_{F,n}[\geq 1]}\Big(e^{h^0(F,\Lambda)}-1\Big)\cdot\mathrm{Vol}(\Lambda)^{1-s} \, d\mu(\Lambda)\\ &\qquad\qquad\qquad +\mathrm{Vol}\Big(\mathcal M_{F,n}[1]\Big)\cdot\Big( \frac{1}{s-1}-\frac{1}{s}\Big)\end{aligned}}}   $\huge{\clubsuit\ \color{Blue}{\mathrm{Zeta\ Facts:}}}$  $\qquad\LARGE{\mathrm{(0)\ (Naturalness)}\LARGE{\qquad \widehat{\zeta}_{F,1}(s)\,\buildrel\cdot\over=\,\widehat\zeta_F(s)\qquad the\ complete\ Dedekind\ zeta\ function;}}$  $\qquad\LARGE{\mathrm{(1)\ (Meromorphic\ Cont)}\qquad\LARGE{\widehat{\zeta}_{F,n}(s)\ admits\ a\ meromorphic\ continuation\ to\ all\ s;}}$  $\qquad\LARGE{\mathrm{ (2)\ (Functional\ Equation)}\hskip 7.0cm \LARGE{\boxed{\widehat{\zeta}_{F,n}(1-s)=\widehat{\zeta}_{F,n}(s)}}}$  $\qquad\LARGE{\mathrm{ (3)\ ({Singularities})}\LARGE{\qquad \widehat{\zeta}_{F,n}(s)\ has\ two\ singularities,\ all\ simple\ poles,\ at\ s=0,1,\ with\ the\ residue}}$ $\LARGE{\boxed{\mathrm{Res}_{s=1}\widehat{\zeta}_{F,n}(s)=\mathrm{Vol}\big({\mathcal M}_{F,n}[1]\big)}}\qquad$