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

Ralph Furmaniak

Abstract

Bohr's last problem (1952) asks whether every ordinary Dirichlet series with nonzero Lindelöf order function $μ$ has $μ'(ω_μ{-}0)\le-1$; a negative answer would imply Lindelöf for $ζ$. Kahane (1989) refuted this with half-plane counterexamples. We study the refinement for series with entire continuation of order $\le 1$: the Analytic Lindelöf Hypothesis that $μ$ is piecewise linear with integer slopes. Deforming the Mellin integral to the strip boundary reduces $μ_L$ to a residue sum over singularities of the generating function on $|x|=1$, giving $μ_L(σ)=\max(0,\tfrac12-σ+ρ)$. For classical $L$-functions this sum is the functional-equation dual, and bounding it is Lindelöf; 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-Lindelöf implies ALH. Our central construction is the Cantor Dirichlet series $L(s)=\sum\hatν(n)n^{-s}$, with $ν$ the ternary Cantor measure. Its Kaczorowski--Perelli twist spectrum is empty; we prove $μ_L(\tfrac12)\le\tfrac18$ unconditionally via a Montgomery--Vaughan argument on the product variable $(m_1+α)(m_2+α)$, where a Vieta identity guarantees distinct frequencies. A Cantor-weighted Hurwitz second-moment conjecture would give $μ_L(\tfrac12)=0$.

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 Lindelöf order function has ; a negative answer would imply Lindelöf for . Kahane (1989) refuted this with half-plane counterexamples. We study the refinement for series with entire continuation of order : the Analytic Lindelöf Hypothesis that is piecewise linear with integer slopes. Deforming the Mellin integral to the strip boundary reduces to a residue sum over singularities of the generating function on , giving . For classical -functions this sum is the functional-equation dual, and bounding it is Lindelöf; 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-Lindelöf implies ALH. Our central construction is the Cantor Dirichlet series , with the ternary Cantor measure. Its Kaczorowski--Perelli twist spectrum is empty; we prove unconditionally via a Montgomery--Vaughan argument on the product variable , where a Vieta identity guarantees distinct frequencies. A Cantor-weighted Hurwitz second-moment conjecture would give .
Paper Structure (44 sections, 41 theorems, 86 equations)

This paper contains 44 sections, 41 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Bohr's problem
  3. Kahane's resolution
  4. The entirety hypothesis and the Analytic Lindelöf Hypothesis
  5. The Lerch reduction
  6. Main results
  7. Paper organization.
  8. Background: Bohr's Summability and the Hankel Contour
  9. Riesz summability and Bohr's sandwich
  10. The Hankel contour
  11. Kahane's Theorem and the 1/2
  12. Kahane's Counterexamples Are Not Entire
  13. Failure of entire continuation
  14. Entire random series have integer slopes
  15. The Refined Sharp Kahane Conjecture
  16. ...and 29 more sections

Key Result

Theorem 1.2

For every ordinary Dirichlet series with analytic continuation in a vertical half-plane, Equivalently, $\mu(\sigma) \ge \max(0,\,\omega_\mu - \tfrac{1}{2} - \sigma)$.

Theorems & Definitions (100)

  • Theorem 1.2: Kahane 1989, Theorem 3.1
  • Conjecture 1.3: Analytic Lindelöf Hypothesis, ALH
  • Theorem 1.4: Lerch-LH ${}\Rightarrow{}$ ALH; partial converse
  • Remark 1.5: Degenerate cases
  • Proposition 2.1
  • Theorem 3.1: Kahane 1989, Thm 3.1
  • Corollary 3.2
  • proof : Proof sketch of Theorem \ref{['thm:kahane-31']}
  • Remark 3.3
  • Remark 3.4: Sharpness
  • ...and 90 more