A zeta-function associated with zero ternary forms

Abstract:

Consider the zeta-function associated with zero ternary forms defined as \[ \tilde \xi (t) = \sum \limits _x {\frac {1} {{{{\left | {\det x} \right |}^t}}}} \quad (\operatorname {Re} t \geqslant 2),\] where $x$ runs over all $\operatorname {SL}_{3}({\mathbf {Z}})$-inequivalent zero ternary forms. We shall approximate $\tilde \xi (t)$ by another zeta-function which we can compute explicitly. By the approximation, we see that $\tilde \xi (2)$ is very close to $2\zeta (2)\zeta (2)$ which gives the contribution of zero ternary forms to the dimension formula of Siegel’s cusp forms of degree three (computing via Selberg Trace Formula) up to a constant multiple.