Explicit Analytic Continuation of Euler Products

TL;DR

This paper explains the 'Factorization Method' for meromorphically continuing Euler products, crucial in arithmetic statistics, providing accessible explanations, existing results, and explicit singularity characterizations.

Contribution

It offers a comprehensive exposition of the Factorization Method, including new insights and detailed proofs for meromorphic continuation of Euler products with constant or Frobenian coefficients.

Findings

Provides a self-contained introduction suitable for new researchers.

Establishes meromorphic continuation of Euler products to Re(s)>0.

Details the locations and orders of all singularities.

Abstract

The generating series of a number of different objects studied in arithmetic statistics can be built out of Euler products. Euler products often have very nice analytic properties, and by constructing a meromorphic continuation one can use complex analytic techniques, including Tauberian theorems to prove asymptotic counting theorems for these objects. One standard technique for producing a meromorphic continuation is to factor out copies of the Riemann zeta function, for which a meromorphic continuation is already known. This paper is an exposition of the "Factorization Method" for meromorphic continuation. We provide the following three resources with an eye towards research in arithmetic statistics: (1) an introduction to this technique targeted at new researchers, (2) exposition of existing works, with self-contained proofs, that give a continuation of Euler products with constant or Frobenain coefficients to the right halfplane ${\rm Re}(s)>0$ (away from an isolated set of singularities), and (3) explicit statements on the locations and orders of all singularities for these Euler products.