Shintani's zeta function is not a finite sum of Euler products

TL;DR

This paper proves that Shintani's zeta function cannot be expressed as a finite sum of Euler products, answering a question posed by Wright negatively and extending the result to related Dirichlet series.

Contribution

It establishes a fundamental limitation on the representation of Shintani's zeta function and related series as finite sums of Euler products.

Findings

Shintani's zeta function cannot be written as a finite sum of Euler products.

The proof extends to several related Dirichlet series.

Answers Wright's question in the negative.

Abstract

We prove that the Shintani zeta function associated to the space of binary cubic forms cannot be written as a finite sum of Euler products. Our proof also extends to several closely related Dirichlet series. This answers a question of Wright in the negative.