Bohr's Last Problem Under the Entirety Hypothesis: A Survey with Initial Reductions

Abstract

Bohr's last problem (1952) asks whether every ordinary Dirichlet series with nonzero Lindel\"of order function $\mu$ has $\mu'(\omega_\mu{-}0)\le-1$; a negative answer would imply Lindel\"of for $\zeta$. Kahane (1989) refuted this with half-plane counterexamples. We study the refinement for series with entire continuation of order $\le 1$: the Analytic Lindel\"of Hypothesis that $\mu$ is piecewise linear with integer slopes. Deforming the Mellin integral to the strip boundary reduces $\mu_L$ to a residue sum over singularities of the generating function on $|x|=1$, giving $\mu_L(\sigma)=\max(0,\tfrac12-\sigma+\rho)$. For classical $L$-functions this sum is the functional-equation dual, and bounding it is Lindel\"of; for self-similar or random singularities it is a Rajchman Fourier transform. We show Kahane's half-plane examples fail entirety, his entire random examples have integer slopes a.s., and Lerch-Lindel\"of implies ALH. Our central construction is the Cantor Dirichlet series $L(s)=\sum\hat\nu(n)n^{-s}$, with $\nu$ the ternary Cantor measure. Its Kaczorowski--Perelli twist spectrum is empty; we prove $\mu_L(\tfrac12)\le\tfrac18$ unconditionally via a Montgomery--Vaughan argument on the product variable $(m_1+\alpha)(m_2+\alpha)$, where a Vieta identity guarantees distinct frequencies. A Cantor-weighted Hurwitz second-moment conjecture would give $\mu_L(\tfrac12)=0$.