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A Note on the Phragmen-Lindelof Theorem

Andrew Fiori

TL;DR

This work generalizes the Phragmén-Lindelöf principle (extending Rademacher) to interpolate holomorphic bounds from boundary lines under refined growth and monotonicity assumptions, while correcting prevalent errors in the literature. It introduces a simple yet flexible main theorem using boundary factors $G_i(s)$ with exponents $\alpha_i,\beta_i$, and explains how to handle negative exponents by a simple multiplier, including standard choices $G(s)=e$ and $G(s)=Q+s$; a midpoint bound is also provided. The paper then develops a more adaptable version to address cases where $|G_i(\sigma+it)|$ is not monotone in $\sigma$, offering strategies to enforce monotonicity and a split-bound variant that treats large and small $t$ separately. It also analyzes errors in prior results, proposing a method using $G_{Q_1,Q_2}(s)=\tfrac{1}{2}\log((Q_1+s)(Q_2-s))$ to recover or improve numerics and extend to iterated logarithms. Finally, it applies the theory to explicit bounds for the Riemann zeta function across several $\sigma$-ranges, constructing concrete bounds with carefully chosen $G_i$ (and, when needed, auxiliary transformations) and demonstrating competitive accuracy against known results for large $|t|$.

Abstract

We provide a generalization of the Phragmén-Lindelöf principal of Rademacher with the aim of correcting, or at least provide a pathway to correcting, several errors appearing in the literature.

A Note on the Phragmen-Lindelof Theorem

TL;DR

This work generalizes the Phragmén-Lindelöf principle (extending Rademacher) to interpolate holomorphic bounds from boundary lines under refined growth and monotonicity assumptions, while correcting prevalent errors in the literature. It introduces a simple yet flexible main theorem using boundary factors with exponents , and explains how to handle negative exponents by a simple multiplier, including standard choices and ; a midpoint bound is also provided. The paper then develops a more adaptable version to address cases where is not monotone in , offering strategies to enforce monotonicity and a split-bound variant that treats large and small separately. It also analyzes errors in prior results, proposing a method using to recover or improve numerics and extend to iterated logarithms. Finally, it applies the theory to explicit bounds for the Riemann zeta function across several -ranges, constructing concrete bounds with carefully chosen (and, when needed, auxiliary transformations) and demonstrating competitive accuracy against known results for large .

Abstract

We provide a generalization of the Phragmén-Lindelöf principal of Rademacher with the aim of correcting, or at least provide a pathway to correcting, several errors appearing in the literature.

Paper Structure

This paper contains 7 sections, 4 theorems, 69 equations.

Table of Contents

  1. Simple form of Main Theorem
  2. Dealing with the Condition of Increasing/Decreasing in $\sigma$
  3. Errors in the literature
  4. Examples
  5. First Example
  6. Second Example
  7. Third Example

Key Result

Theorem 1

Let $a$ and $b$ be real numbers and let $G_1,\ldots, G_r$ be complex functions that are holomorphic for ${\rm Re}(s)\in [a,b]$, have $\overline{G_i(s)}=G_i(\overline{s})$, and satisfy the following growth condition for all $\sigma\in[a,b]$ for some $C_1,C_2>0$ and $0<C_3<\frac{\pi}{b-a}$. Suppose that $f$ is a holomorphic function for ${\rm Re}(s)\in [a,b]$ and satisfies the growth condition for

Theorems & Definitions (12)

  • Theorem 1: Phragmén-Lindelöf principle
  • Remark 2
  • Corollary 3
  • proof
  • Remark 4
  • Lemma 5
  • proof
  • proof : Proof of Theorem \ref{['thm:phrag']}
  • Remark 6
  • Theorem 7
  • ...and 2 more