Date Approved

2024

Degree Type

Open Access Senior Honors Thesis

Department or School

Mathematics and Statistics

First Advisor

Brandon Alberts, Ph.D.

Second Advisor

Andrew Ross, Ph.D.

Third Advisor

Debra Ingram, Ph.D.

Abstract

In this paper, we investigate the meromorphic continuation of Euler products, with a particular focus on those with polynomial growth coefficients. Building upon the classical example of the Riemann zeta function, we derive conditions under which Euler products can be extended beyond their regions of convergence to broader domains of the complex plane, revealing singularities and deepening our understanding of their analytic properties. We begin by establishing a framework for meromorphic continuation and then introduce a general theorem that applies to Euler products of the form

where the coefficients and exponents satisfy specific constraints. We prove that such Euler products converge absolutely in certain regions and possess meromorphic continuations to open neighborhoods of these regions, with singularities characterized by the maximum of certain critical exponents. The Factorization Method, which involves factoring out copies of the Riemann zeta function, is employed to prove the meromorphic continuation results.

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Mathematics Commons

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