<p>For any <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$0\le \sigma \le 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>≤</mo> <mi>σ</mi> <mo>≤</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and any <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$T>10$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>></mo> <mn>10</mn> </mrow> </math></EquationSource> </InlineEquation> sufficiently large, let <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$N_{\zeta }(\sigma ,K,T)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>N</mi> <mi>ζ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>σ</mi> <mo>,</mo> <mi>K</mi> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the number of zeros <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\rho =\beta +i\gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ρ</mi> <mo>=</mo> <mi>β</mi> <mo>+</mo> <mi>i</mi> <mi>γ</mi> </mrow> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\zeta _{K}(s)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ζ</mi> <mi>K</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$|\gamma |\le T$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>γ</mi> <mo stretchy="false">|</mo> <mo>≤</mo> <mi>T</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\beta \ge \sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo>≥</mo> <mi>σ</mi> </mrow> </math></EquationSource> </InlineEquation> and the zero being counted according  to multiplicity. For <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$k\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, we have <Equation ID="Equ14"> <EquationSource Format="TEX">$ N_{\zeta }(\sigma ,K,T)\ll T^{\frac{2k}{6\sigma -3}(1-\sigma )+\varepsilon }, $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mi>N</mi> <mi>ζ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>σ</mi> <mo>,</mo> <mi>K</mi> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mo>≪</mo> <msup> <mi>T</mi> <mrow> <mfrac> <mrow> <mn>2</mn> <mi>k</mi> </mrow> <mrow> <mn>6</mn> <mi>σ</mi> <mo>-</mo> <mn>3</mn> </mrow> </mfrac> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <mi>σ</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>ε</mi> </mrow> </msup> <mo>,</mo> </mrow> </math></EquationSource> </Equation>where <Equation ID="Equ15"> <EquationSource Format="TEX">$ \frac{2k+3}{2k+6}\le \sigma <1 $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mfrac> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>3</mn> </mrow> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>6</mn> </mrow> </mfrac> <mo>≤</mo> <mi>σ</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </Equation>and the implied constant may depend on the number field <i>K</i> and <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\varepsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation>. This improves previous results for <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$k\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> of certain range of <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>.</p>

On density of the zeros of Dedekind zeta-functions

期刊论文
发表日期：

2026年2月1日

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Wei Zhang

摘要

For any $\sigma $ $σ$ with $0\le \sigma \le 1$ $0 \leq σ \leq 1$ and any $T>10$ $T > 10$ sufficiently large, let $N_{\zeta }(\sigma ,K,T)$ $N_{ζ} (σ, K, T)$ be the number of zeros $\rho =\beta +i\gamma $ $ρ = β + i γ$ of $\zeta _{K}(s)$ $ζ_{K} (s)$ with $|\gamma |\le T$ $| γ | \leq T$ and $\beta \ge \sigma $ $β \geq σ$ and the zero being counted according to multiplicity. For $k\ge 3$ $k \geq 3$ , we have $ N_{\zeta }(\sigma ,K,T)\ll T^{\frac{2k}{6\sigma -3}(1-\sigma )+\varepsilon }, $ $N_{ζ} (σ, K, T) ≪ T^{\frac{2 k}{6 σ - 3} (1 - σ) + ε},$ where $ \frac{2k+3}{2k+6}\le \sigma <1 $ $\frac{2 k + 3}{2 k + 6} \leq σ < 1$ and the implied constant may depend on the number field K and $\varepsilon $ $ε$ . This improves previous results for $k\ge 3$ $k \geq 3$ of certain range of $\sigma $ $σ$ .