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<p>We investigate the distribution of large values of the Riemann zeta function <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1955_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="32" /> </InlineMediaObject> <EquationSource Format="TEX">\(\zeta (s)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> in the strip <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1955_Article_IEq5.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="93" /> </InlineMediaObject> <EquationSource Format="TEX">\(\frac{1}{2}&lt;{\Re e\,}s&lt;1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>&lt;</mo> <mrow> <mi>ℜ</mi> <mi>e</mi> <mspace width="0.166667em" /> </mrow> <mi>s</mi> <mo>&lt;</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. For any fixed <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1955_Article_IEq6.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="122" /> </InlineMediaObject> <EquationSource Format="TEX">\({\Re e\,}s=\sigma \in (\frac{1}{2},1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi>ℜ</mi> <mi>e</mi> <mspace width="0.166667em" /> </mrow> <mi>s</mi> <mo>=</mo> <mi>σ</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, we obtain an improved distribution function of large values of <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1955_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">\(|\zeta (\sigma +\textrm{i}t)|\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>σ</mi> <mo>+</mo> <mtext>i</mtext> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">|</mo> </mrow> </math></EquationSource> </InlineEquation>, holding in the same range as that given by Lamzouri.</p>

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On the Distribution of large values of \(|\zeta (\sigma +\textrm{i}t)|\)

  • 期刊论文

  • 发表日期：

    2025年8月26日

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  • Zikang Dong

摘要

We investigate the distribution of large values of the Riemann zeta function \(\zeta (s)\) ζ ( s ) in the strip \(\frac{1}{2}<{\Re e\,}s<1\) 1 2 < ℜ e s < 1 . For any fixed \({\Re e\,}s=\sigma \in (\frac{1}{2},1)\) ℜ e s = σ ∈ ( 1 2 , 1 ) , we obtain an improved distribution function of large values of \(|\zeta (\sigma +\textrm{i}t)|\) | ζ ( σ + i t ) | , holding in the same range as that given by Lamzouri.