<p>In this paper we study an <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$L^{p}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation> analogue of Bohr’s abscissae of summability for Dirichlet series. For polynomially bounded analytic functions in a strip with order function <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation>, convexity of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$1/\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mi>μ</mi> </mrow> </math></EquationSource> </InlineEquation> is equivalent to approximate concavity of the abscissae in <i>p</i>. If <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> obeys a functional equation of the Selberg class type, this is equivalent to the Lindelöf hypothesis if <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mu '(1/2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>μ</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> does not exist. Otherwise, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> is everywhere differentiable (therefore subconvex) with quadratic decay near one.</p>

On the Phragmen-Lindelöf theorem in strips

期刊论文
发表日期：

2026年7月27日

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Kevin Smith

摘要

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