<p>For integers <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$m,m' \geqslant 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>,</mo> <msup> <mi>m</mi> <mo>′</mo> </msup> <mo>⩾</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\pi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>π</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\pi '$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>π</mi> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation> be cuspidal automorphic representations of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\textrm{GL}(m)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>GL</mtext> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\textrm{GL}(m')$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>GL</mtext> <mo stretchy="false">(</mo> <msup> <mi>m</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, respectively. We present a new proof of zero-free regions for <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$L(s, \pi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>,</mo> <mi>π</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$L(s, \pi \hspace{1.111pt}{\times }\hspace{1.111pt}\pi ')$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>,</mo> <mi>π</mi> <mspace width="1.111pt" /> <mo>×</mo> <mspace width="1.111pt" /> <msup> <mi>π</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> under the assumption that <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\pi , \pi '$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>π</mi> <mo>,</mo> <msup> <mi>π</mi> <mo>′</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$L(s,\pi \hspace{1.111pt}{\times }\hspace{1.111pt}\pi ')$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>,</mo> <mi>π</mi> <mspace width="1.111pt" /> <mo>×</mo> <mspace width="1.111pt" /> <msup> <mi>π</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is self-dual. Our approach builds on ideas of “pretentious” multiplicative functions due to Granville and Soundararajan (as presented by Koukoulopoulos) and the notion of a positive semi-definite family of automorphic <i>L</i>-functions due to Lichtman and Pascadi.</p>

A metric approach to zero-free regions for L-functions

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2026年7月6日

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Nawapan Wattanawanichkul

摘要

For integers $m,m' \geqslant 1$ $m, m^{'} ⩾ 1$ , let $\pi $ $π$ and $\pi '$ $π^{'}$ be cuspidal automorphic representations of $\textrm{GL}(m)$ $GL (m)$ and $\textrm{GL}(m')$ $GL (m^{'})$ , respectively. We present a new proof of zero-free regions for $L(s, \pi )$ $L (s, π)$ and for $L(s, \pi \hspace{1.111pt}{\times }\hspace{1.111pt}\pi ')$ $L (s, π \times π^{'})$ under the assumption that $\pi , \pi '$ $π, π^{'}$ or $L(s,\pi \hspace{1.111pt}{\times }\hspace{1.111pt}\pi ')$ $L (s, π \times π^{'})$ is self-dual. Our approach builds on ideas of “pretentious” multiplicative functions due to Granville and Soundararajan (as presented by Koukoulopoulos) and the notion of a positive semi-definite family of automorphic L-functions due to Lichtman and Pascadi.