Computation of Weng's rank 2 zeta function over an algebraic number field
Weng's rank \(r\) non-abelian zeta-function \(\xi_{F,r}\) of an algebraic number field \(F\) is represented as an integral of the Eisenstein series over the moduli space of the semistable \(O_F\)-lattices with rank \(r\). The author proves that \[ \xi_{F,2}(s)=C_0\left(\frac{\widehat{\zeta_F}(2s)}{s-1}-\frac{\widehat{\zeta_F}(2s-1)}{s}\right), \] where \( \widehat{\zeta_F}(s)\) is the completed Dedekind zeta function of \(F\) and \(C_0\) is a non-zero constant. Moreover, \(\xi_{F,2}(s)\) has all its zeros on the critical line \(\Re(s)=\frac12.\)
scientific article; zbMATH DE number 5190754
scientific article