On Certain Multiple Dirichlet Series

In 2013, Li-Mei Lim generalized the result of Chinta and Offen on Orthogonal Period of a $\operatorname{GL}_3$ Eisenstein series to the case of the minimal parabolic $\operatorname{GL}_3$ Eisenstein series using multiple Dirichlet series associated to the prehomogeneous vector space of ternary quadratic forms. In this thesis, we show similar results using the prehomogeneous vector space of binary cubic forms. Under the action of $\Gamma_\infty$ on to the prehomogeneous vector space of "positive definite" binary cubic forms, there are four invariants, and using two out of four invariants, we set up a multiple Dirichlet series. With reduction theory, local Euler factor computation, a functional equation and convexity arguments, we prove that this multiple Dirichlet series can be meromorphically continued to the whole $\mathbb{C}^2$.