[Paper Review] Essential singularities of Euler products

Gautami Bhowmik, Jan‐Christoph Schlage‐Puchta|arXiv (Cornell University)|Jan 12, 2010

Advanced Mathematical Identities9 references3 citations

TL;DR

This paper classifies essential singularities of Dirichlet series with Euler products over rational functions in $ p $ and $ p^{-s} $, proving that natural boundaries arise when the main part of the Euler product is non-cyclotomic or when certain local zeros accumulate near $ \Re s = \beta $. The key contribution is a complete five-case classification of meromorphic continuation behavior based on the structure of the polynomial $ W(X,Y) $, showing that Estermann’s method fails in general and that multivariable zeta functions are technically simpler to analyze than their 1.5-variable counterparts.

ABSTRACT

We classify singularities of Dirichlet series having Euler products which are rational functions for p and p^{-s} for p a prime number and give examples of natural boundaries from zeta functions of groups and height zeta functions.

Motivation & Objective

• To classify the singularities of Dirichlet series with Euler products of the form $ D(s) = \prod_p W(p, p^{-s}) $, where $ W $ is a rational function.
• To determine conditions under which such series admit meromorphic continuation beyond $ \Re s = \beta $, or develop a natural boundary at $ \Re s = \beta $.
• To demonstrate that Estermann’s classical method of pole/zero accumulation fails to prove the widely believed conjecture on meromorphic continuation.
• To compare the complexity of the 1.5-variable problem (one complex variable, two-variable polynomial) with the full multivariable case.
• To show that multivariable zeta functions $ D(s_1, s_2) = \prod_p W(p^{-s_1}, p^{-s_2}) $ are easier to analyze and have a complete classification of natural boundaries.

Proposed method

• Define $ \alpha = \sup\left\{ \frac{n+1}{m} : a_{n,m} \neq 0 \right\} $ and $ \beta = \sup\left\{ \frac{n}{m} : a_{n,m} \neq 0 \right\} $, which determine the convergence and continuation domains.
• Introduce the 'main part' $ \tilde{W} $ of $ W $, consisting only of terms achieving $ \frac{n}{m} = \beta $, which governs the analytic behavior of the Euler product.
• Classify the behavior of $ D(s) $ into five cases based on whether $ \tilde{W} $ is cyclotomic, whether $ W \neq \tilde{W} $, and whether local zeros of $ W(p, p^{-s}) $ accumulate to the right of $ \Re s = \beta $.
• Use the prime number theorem in short intervals to show that if $ \tilde{W} $ is non-cyclotomic, then the set of primes for which $ W(p, p^{-s}) $ has a zero with $ \Re s > \beta $ is infinite.
• Apply results from multivariable zeta functions, particularly the theorem that $ D(s_1, s_2) $ is meromorphically continuable to $ \mathbb{C}^2 $ if and only if $ W $ is cyclotomic.
• Construct counterexamples to show that Estermann’s method—based on accumulation of poles or zeros on $ \Re s = \beta $—cannot capture all cases, especially when zeros of $ \zeta(s) $ interfere.

Experimental results

Research questions

• RQ1 Under what conditions does a Dirichlet series $ D(s) = \prod_p W(p, p^{-s}) $ with rational $ W $ admit meromorphic continuation beyond $ \Re s = \beta $?
• RQ2 Why does Estermann’s method of pole/zero accumulation fail to prove the conjecture that only finite products of $ \zeta $-functions are continuable?
• RQ3 How does the behavior of the 1.5-variable Euler product compare to the full multivariable zeta function $ D(s_1, s_2) = \prod_p W(p^{-s_1}, p^{-s_2}) $?
• RQ4 What role do zeros of the Riemann zeta function play in obstructing meromorphic continuation of $ D(s) $, and in which cases can they be controlled?
• RQ5 Can the natural boundary of $ D(s) $ be fully characterized by the cyclotomic nature of the main part $ \tilde{W} $, and what are the exceptions?

Key findings

• The Euler product $ D(s) = \prod_p W(p, p^{-s}) $ converges in the half-plane $ \Re s > \alpha $, and admits meromorphic continuation to $ \Re s > \beta $, where $ \alpha $ and $ \beta $ are defined via the exponents of $ W(X,Y) $.
• Case (1): If $ W $ is cyclotomic and $ W = \tilde{W} $ after removing unitary factors, then $ D(s) $ is a finite product of Riemann zeta functions and is meromorphically continuable to $ \mathbb{C} $.
• Case (2): If $ \tilde{W} $ is not cyclotomic, then every point on the line $ \Re s = \beta $ is an obstructing point, meaning it is an essential singularity due to accumulation of poles or zeros.
• Case (3): If $ W \neq \tilde{W} $, $ \tilde{W} $ is cyclotomic, and there are infinitely many $ (n,m) $ with $ \frac{n}{m} < \beta < \frac{n+1}{m} $, then $ \beta $ is an obstructing point.
• Case (4): If $ W \neq \tilde{W} $, $ \tilde{W} $ is cyclotomic, only finitely many $ (n,m) $ satisfy $ \frac{n}{m} < \beta < \frac{n+1}{m} $, but infinitely many primes $ p $ yield zeros of $ W(p, p^{-s}) $ with $ \Re s_0 > \beta $, then every point on $ \Re s = \beta $ is an obstructing point.
• Case (5): If none of the above, then no point on $ \Re s = \beta $ is an obstructing point, and the behavior is not detectable by pole/zero accumulation alone—this case is not captured by Estermann’s method.