Assuming the Generalized Riemann Hypothesis (GRH), we utilize the long resonator method to derive <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3187_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation>-results for the family of quadratic Dirichlet L-functions <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3187_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">$L(\sigma , \chi _d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mo stretchy="false">(</mo> <mi>σ</mi> <mo>,</mo> <msub> <mi>χ</mi> <mi>d</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where d runs over all fundamental discriminants with <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3187_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="61" /> </InlineMediaObject> <EquationSource Format="TEX">$|d| \le X$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>d</mi> <mo stretchy="false">|</mo> <mo>≤</mo> <mi>X</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3187_Article_IEq4.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">$\sigma \in \left[ \frac{1}{2}, 1 \right] $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>σ</mi> <mo>∈</mo> <mfenced close="]" open="["> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>,</mo> <mn>1</mn> </mfenced> </mrow> </math></EquationSource> </InlineEquation> is fixed. This study advances understanding of the maximum size of <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3187_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">$L(\sigma , \chi _d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mo stretchy="false">(</mo> <mi>σ</mi> <mo>,</mo> <msub> <mi>χ</mi> <mi>d</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> within the segment <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3187_Article_IEq4.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">$\sigma \in \left[ \frac{1}{2}, 1 \right] $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>σ</mi> <mo>∈</mo> <mfenced close="]" open="["> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>,</mo> <mn>1</mn> </mfenced> </mrow> </math></EquationSource> </InlineEquation>. In particular, we improve upon Soundararajan’s results at the central point and provide a lower bound on the proportion of fundamental discriminants, uniformly within an expected order of magnitude, up to optimal values of the constant for a fixed <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3187_Article_IEq7.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="72" /> </InlineMediaObject> <EquationSource Format="TEX">$\sigma \in \left( \frac{1}{2}, 1 \right] $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>σ</mi> <mo>∈</mo> <mfenced close="]" open="("> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>,</mo> <mn>1</mn> </mfenced> </mrow> </math></EquationSource> </InlineEquation>.

Large values of quadratic Dirichlet L-functions

期刊论文
发表日期：

2025年6月4日

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Pranendu Darbar,
Gopal Maiti

摘要

Assuming the Generalized Riemann Hypothesis (GRH), we utilize the long resonator method to derive $\Omega $ $Ω$ -results for the family of quadratic Dirichlet L-functions $L(\sigma , \chi _d)$ $L (σ, χ_{d})$ , where d runs over all fundamental discriminants with $|d| \le X$ $| d | \leq X$ and $\sigma \in \left[ \frac{1}{2}, 1 \right] $ $σ \in [\frac{1}{2}, 1]$ is fixed. This study advances understanding of the maximum size of $L(\sigma , \chi _d)$ $L (σ, χ_{d})$ within the segment $\sigma \in \left[ \frac{1}{2}, 1 \right] $ $σ \in [\frac{1}{2}, 1]$ . In particular, we improve upon Soundararajan’s results at the central point and provide a lower bound on the proportion of fundamental discriminants, uniformly within an expected order of magnitude, up to optimal values of the constant for a fixed $\sigma \in \left( \frac{1}{2}, 1 \right] $ $σ \in (\frac{1}{2}, 1]$ .