On the Phragmen Lindel\"of Theorem in strips

TL;DR

This paper explores an $L^{p}$ analogue of Bohr's abscissae for Dirichlet series, linking convexity of the order function to properties of the abscissae and relating to the Lindel"of hypothesis.

Contribution

It establishes a connection between the convexity of the order function and the behavior of abscissae in an $L^{p}$ setting, extending classical results.

Findings

Convexity of 1/μ is equivalent to approximate concavity of abscissae in p.

If μ satisfies a Selberg-type functional equation, this relates to the Lindel"of hypothesis.

μ's differentiability and decay properties are characterized based on its behavior near 1.

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.