A two variable zeta function associated to the space of binary forms of degree $d$

TL;DR

This paper proves that a two-variable zeta function, associated with binary forms of degree d, can be extended analytically across the entire complex plane as a meromorphic function.

Contribution

It establishes the analytic continuation of a new two-variable zeta function linked to binary forms, expanding understanding of its complex analytic properties.

Findings

Zeta function extends meromorphically to the entire complex plane.

Analytic continuation is achieved for the two-variable zeta function.

The work advances the theory of zeta functions related to algebraic forms.

Abstract

In this paper we prove the analytic continuation of a two variable zeta function defined using the vector space of binary forms of degree $d$ to the entire two dimensional complex space as a meromorphic function.