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<p>Assuming the Generalized Riemann Hypothesis and a pair correlation conjecture for the zeros of Dirichlet <i>L</i>-functions, we establish the truth of a conjecture of Montgomery (in its corrected form stated by Friedlander and Granville) on the magnitude of the error term in the prime number theorem in arithmetic progressions. As a consequence, we obtain that, under the same assumptions, the Elliott–Halberstam conjecture holds true. As another consequence, under the same assumptions, we will show that the number of Dirichlet characters <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\chi (\text {mod}\,q)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>χ</mi> <mo stretchy="false">(</mo> <mtext>mod</mtext> <mspace width="0.166667em" /> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for which <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(L(\frac{1}{2},\chi )=0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mo stretchy="false">(</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>,</mo> <mi>χ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> is of order less than <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(q^{1/2+\varepsilon }\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>q</mi> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> <mo>+</mo> <mi>ε</mi> </mrow> </msup> </math></EquationSource> </InlineEquation>.</p>

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Pair correlation of zeros of Dirichlet L-functions: a possible path towards the conjectures of Chowla, Elliott-Halberstam and Montgomery

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  • 发表日期：

    2026年2月13日

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  • Neelam Kandhil,
  • Alessandro Languasco,
  • Pieter Moree

摘要

Assuming the Generalized Riemann Hypothesis and a pair correlation conjecture for the zeros of Dirichlet L-functions, we establish the truth of a conjecture of Montgomery (in its corrected form stated by Friedlander and Granville) on the magnitude of the error term in the prime number theorem in arithmetic progressions. As a consequence, we obtain that, under the same assumptions, the Elliott–Halberstam conjecture holds true. As another consequence, under the same assumptions, we will show that the number of Dirichlet characters \(\chi (\text {mod}\,q)\) χ ( mod q ) for which \(L(\frac{1}{2},\chi )=0\) L ( 1 2 , χ ) = 0 is of order less than \(q^{1/2+\varepsilon }\) q 1 / 2 + ε .