The Analytic Lindelöf Hypothesis for ordinary Dirichlet series

Let $L(s)=\sum a_n n^{-s}$ be an ordinary Dirichlet series with entire continuation of order at most $1$, or meromorphic continuation with finitely many poles. Define the order function $\mu(\sigma)$ from the vertical growth of $L$ on the line $\operatorname{Re}(s)=\sigma$. Analytic Lindelöf Hypothesis. The function $\mu(\sigma)$ is piecewise linear with slopes in $\mathbb{Z}_{\le 0}$. This refines Bohr's problem under the entirety hypothesis; the paper presents it as an open hypothesis, motivated by the absence of known entire ordinary Dirichlet-series examples with nonintegral slopes.