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<p>Landau–Siegel zeros are hypothetical zeros of Dirichlet <i>L</i>-functions that are close to the point <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(s=1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. A classic theorem of Page shows at most one such zero can exist among all Dirichlet <i>L</i>-functions with conductor <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\le Q\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>≤</mo> <mi>Q</mi> </mrow> </math></EquationSource> </InlineEquation>. We show that one can significantly refine Page’s theorem under the assumption that all non-real zeros of Dirichlet <i>L</i>-functions lie outside a shrinking neighborhood of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(s=1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

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A Conditional Refinement of Page’s Theorem on zeros of Dirichlet L-functions

  • 期刊论文

  • 发表日期：

    2026年2月18日

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  • Debmalya Basak,
  • Kyle Pratt

摘要

Landau–Siegel zeros are hypothetical zeros of Dirichlet L-functions that are close to the point \(s=1\) s = 1 . A classic theorem of Page shows at most one such zero can exist among all Dirichlet L-functions with conductor \(\le Q\) ≤ Q . We show that one can significantly refine Page’s theorem under the assumption that all non-real zeros of Dirichlet L-functions lie outside a shrinking neighborhood of \(s=1\) s = 1 .