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

TL;DR

This survey explores Bohr's last problem and the Analytic Lindel"of Hypothesis, analyzing Dirichlet series with entire continuation, and presents new results on the Cantor Dirichlet series and its properties.

Contribution

It provides a comprehensive survey of Bohr's last problem under the Entirety Hypothesis, introduces new constructions like the Cantor Dirichlet series, and establishes bounds on its Lindel"of order.

Findings

Kahane's half-plane examples fail entirety

Cantor Dirichlet series has an unconditionally proven Lindel"of order ≤ 1/8

A Cantor-weighted Hurwitz second-moment conjecture could imply zero Lindel"of order at 1/2

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 $\mu_L(\tfrac12)=0$.