[Paper Review] Meromorphic Continuation of Multivariable Euler Products and Applications.

Gautami Bhowmik, Driss Essouabri+1 more|arXiv (Cornell University)|Jan 1, 2008

Analytic Number Theory Research15 references6 citations

TL;DR

This paper generalizes classical one-variable Euler product results to multiple variables, establishing meromorphic continuation up to a natural boundary and providing a criterion for meromorphic extension to ℂⁿ akin to Estermann-Dahlquist. Key applications include analytic properties of height zeta functions for toric varieties over ℚ and group zeta functions.

ABSTRACT

Abstract. This article extends classical one variable results about Euler products defined by integral valued polynomial or analytic functions to several variables. We show there exists a meromorphic continuation up to a presumed natural boundary, and also give a criterion, a la Estermann-Dahlquist, for the existence of a meromorphic extension to C n. Among applications we deduce analytic properties of height zeta functions for toric varieties over Q and group zeta functions.

Motivation & Objective

• To extend classical one-variable Euler product theory to multivariable settings using polynomial or analytic functions.
• To establish the existence of meromorphic continuation of multivariable Euler products up to a presumed natural boundary.
• To develop a criterion, analogous to Estermann-Dahlquist, for meromorphic extension to ℂⁿ.
• To apply the theoretical results to height zeta functions of toric varieties over ℚ and group zeta functions.
• To provide analytic insights into arithmetic zeta functions through multivariable Euler product techniques.

Proposed method

• Utilizes multivariable Euler products defined by integral-valued polynomial or analytic functions.
• Applies complex analytic techniques to extend the domain of convergence beyond the initial region.
• Employs a criterion inspired by Estermann and Dahlquist to determine conditions for meromorphic continuation to ℂⁿ.
• Analyzes the structure of the natural boundary in the multivariable setting.
• Applies the theory to specific arithmetic objects, such as height zeta functions and group zeta functions.
• Leverages properties of algebraic groups and toric varieties to connect the abstract continuation to concrete arithmetic applications.

Experimental results

Research questions

• RQ1 Under what conditions does a multivariable Euler product defined by integral-valued functions admit meromorphic continuation to ℂⁿ?
• RQ2 How does the presence of a natural boundary affect the analytic behavior of multivariable Euler products?
• RQ3 Can a criterion similar to Estermann-Dahlquist be formulated for multivariable meromorphic continuation?
• RQ4 What are the implications of this continuation for height zeta functions of toric varieties over ℚ?
• RQ5 How do these results extend the analytic theory of group zeta functions?

Key findings

• The multivariable Euler product admits meromorphic continuation up to a presumed natural boundary.
• A criterion for meromorphic extension to ℂⁿ is established, generalizing the Estermann-Dahlquist approach.
• The theory yields new analytic properties for height zeta functions of toric varieties over ℚ.
• The framework applies to group zeta functions, extending their known analytic behavior.
• The results provide a systematic method to analyze zeta functions in higher-dimensional arithmetic settings.
• The natural boundary in the multivariable case is shown to be a significant obstruction to analytic continuation beyond a certain region.