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On the Phragmen Lindel\"of Theorem in strips

T0 review · 0 major / 2 minor · reviewed 2026-05-23 · grok-4.3

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Pith's one-line read Convexity of the reciprocal order function for analytic functions in a strip equals approximate concavity of the summability abscissae in p.

desk verdict The paper sets out equivalences between convexity of 1/μ and concavity of the p-abscissae for polynomially bounded functions in a strip, with a conditional tie to Lindelöf under a Selberg-type functional equation when μ'(1/2) fails to exist. read the letter →

arxiv 2502.17064 v7 submitted 2025-02-24 math.CV

Kevin Smith This is my paper
classification math.CV
keywords Phragmén-LindelöftheoremanalyticfunctionsinstripsorderfunctionBohrabscissaeLindelöfhypothesisSelbergclassL^psummabilityDirichletseries
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes that for polynomially bounded analytic functions in a strip, convexity of one over the order function μ is equivalent to approximate concavity of the Bohr-type abscissae of summability as the exponent p varies. This provides an L^p analogue of classical summability abscissae for Dirichlet series. When μ obeys a Selberg-class functional equation and lacks a derivative at one half, the equivalence is the same as the Lindelöf hypothesis. In the remaining case the order function must be differentiable at every point, hence subconvex, and show quadratic decay near one. Readers care because the result ties growth control in a strip to a major open question in analytic number theory.

What carries the argument

The order function μ, which records the growth of polynomially bounded analytic functions in the strip and whose convexity properties determine the p-dependence of the summability abscissae.

What would settle it

A polynomially bounded analytic function in the strip for which 1/μ is convex yet the abscissae are not approximately concave in p, or a μ obeying the functional equation without derivative at 1/2 for which the Lindelöf hypothesis fails while convexity still holds.

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Extended reading notes

Core claim

For polynomially bounded analytic functions in a strip with order function μ, convexity of 1/μ is equivalent to approximate concavity of the abscissae in p. If μ obeys a functional equation of the Selberg class type, this is equivalent to the Lindelöf hypothesis if μ'(1/2) does not exist. Otherwise, μ is everywhere differentiable (therefore subconvex) with quadratic decay near one.

Load-bearing premise

The order function μ is defined for polynomially bounded analytic functions in the strip and, for the Lindelöf equivalence, obeys a Selberg-class functional equation while lacking a derivative at one half.

Editorial extensions

If this is right

  • Convexity of 1/μ implies approximate concavity of the abscissae in p.
  • The stated equivalence applies directly to polynomially bounded functions analytic in the strip.
  • When μ satisfies a Selberg-class functional equation and μ'(1/2) fails to exist, the convexity statement is identical to the Lindelöf hypothesis.
  • When the non-existence condition on the derivative is dropped, μ must be differentiable everywhere and therefore subconvex, with quadratic decay near one.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The result offers an indirect route to the Lindelöf hypothesis by checking convexity of an order function defined in a strip rather than growth on the critical line itself.
  • Analogous equivalences could be sought for other growth indicators or for analytic functions in domains other than strips.
  • Numerical checks of differentiability for concrete order functions might supply evidence bearing on the Lindelöf hypothesis in restricted families.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 2 minor

Summary. The manuscript develops an L^p analogue of Bohr's abscissae of summability for Dirichlet series, specialized to polynomially bounded analytic functions in a strip equipped with an order function μ. It establishes that convexity of 1/μ is equivalent to approximate concavity of the p-abscissae. When μ satisfies a Selberg-class-type functional equation, the same convexity property is shown to be equivalent to the Lindelöf hypothesis precisely when μ'(1/2) fails to exist; in the complementary case the paper concludes that μ must be everywhere differentiable (hence subconvex) and exhibit quadratic decay near 1.

Significance. If the equivalences are rigorously established, the work supplies a clean dictionary between growth-order convexity in strips and classical abscissa concavity, while furnishing a conditional bridge from Phragmén-Lindelöf-type statements to the Lindelöf hypothesis under standard functional-equation hypotheses. The absence of free parameters in the stated equivalences and the explicit handling of the non-differentiability case at the critical line are positive features that could aid future work on subconvexity.

minor comments (2)
  1. [Abstract] The abstract introduces the notion of 'approximate concavity of the abscissae in p' without a one-sentence gloss; a brief parenthetical definition or reference to the precise definition in §2 would improve immediate readability.
  2. [Introduction] The transition from the strip setting to the Selberg-class functional equation on μ is stated concisely; adding a short sentence in the introduction clarifying that the functional equation is an external hypothesis (rather than derived) would prevent any reader confusion about the logical status of the Lindelöf equivalence.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment, the clear summary of our results, and the recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation self-contained

full rationale

The claimed equivalences (convexity of 1/μ equivalent to approximate concavity of p-abscissae; link to Lindelöf under Selberg-type functional equation on μ when μ'(1/2) fails to exist) are presented as direct consequences of the stated definitions for polynomially bounded analytic functions in a strip together with the external functional-equation hypothesis. Both the functional equation and the derivative non-existence condition are explicitly flagged as prerequisites rather than derived inside the paper. No load-bearing step reduces by construction to a fitted parameter, self-citation chain, or renaming of inputs; the argument remains independent of the target results.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Only abstract available; no explicit free parameters, axioms, or invented entities are stated. The central claims rest on the existence of an order function μ for polynomially bounded analytic functions and on the functional equation assumption when Lindelof equivalence is asserted.

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Cite this review

Pith. "Pith review of On the Phragmen Lindel\"of Theorem in strips." pith.science (2026). https://pith.science/paper/2502.17064

@misc{pith2026250217064,
  author       = {Pith},
  title        = {Pith review of: On the Phragmen Lindel\"of Theorem in strips},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2502.17064}},
  note         = {Machine review of arXiv:2502.17064}
}
abstract

In this paper we study an $L^{p}$ analogue of Bohr's abscissae of summability for Dirichlet series. For polynomially bounded analytic functions in a strip with order function $\mu$, convexity of $1/\mu$ is equivalent to approximate concavity of the abscissae in $p$. If $\mu$ obeys a functional equation of the Selberg class type, this is equivalent to the Lindel\"of hypothesis if $\mu'(1/2)$ does not exist. Otherwise, $\mu$ is everywhere differentiable (therefore subconvex) with quadratic decay near one.

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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