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 Frobenian coefficients
to the right half-plane
(away from an isolated set of singularities); and (3) explicit statements on
the locations and orders of all singularities for these Euler products.
