Rudnick–du Sautoy conjecture on meromorphic continuation of Euler products

Let $h(X_1,X_2)\in\mathbb{Z}[X_1,X_2]$ and define

$$Z(s)=\prod_{p\\ \ \textrm{prime}}h(p^{-s},p).$$

A polynomial $g(U)$ is cyclotomic here if it divides $(1-U^{m})^{n}$ for some positive integers $m$ and $n$. Rudnick–du Sautoy conjecture. The function $Z(s)$ can be meromorphically continued to the whole complex plane if and only if there exist cyclotomic polynomials $g_i(U)$, for $i=1,\dots,m$, and integers $u_i,v_i$ such that

$$h(X_1,X_2)=g_1(X_1^{u_1}X_2^{v_1})\cdots g_m(X_1^{u_m}X_2^{v_m}).$$

This conjecture characterizes precisely the Euler products of this type that have no natural boundary. It is presented as a prediction of the maximal domain of meromorphy and is the basis for the subsequent natural-boundary statement.