New Zero-Density Results for Automorphic <i>L</i>-Functions of <i>GL</i>(<i>n</i>)

• Main Authors: Wenjing Ding, Huafeng Liu, Deyu Zhang
• Format: Article
• Language: English
• Published: MDPI AG 2021-08-01
• Series: Mathematics
• Subjects: zero density; automorphic L-function; automorphic representation
• Online Access: https://www.mdpi.com/2227-7390/9/17/2061

Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mo>(</mo><mi>s</mi><mo>,</mo><mi>π</mi><mo>)</mo></mrow></semantics></math></inline-formula> be an automorphic <i>L</i>-function of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mi>L</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>π</mi></semantics></math></inline-formula> is an automorphic representation of group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mi>L</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></semantics></math></inline-formula> over rational number field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">Q</mi></semantics></math></inline-formula>. In this paper, we study the zero-density estimates for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mo>(</mo><mi>s</mi><mo>,</mo><mi>π</mi><mo>)</mo></mrow></semantics></math></inline-formula>. Define <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>N</mi><mi>π</mi></msub><mrow><mo>(</mo><mi>σ</mi><mo>,</mo><msub><mi>T</mi><mn>1</mn></msub><mo>,</mo><msub><mi>T</mi><mn>2</mn></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula> = ♯ {<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula> = <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> + <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>i</mi><mi>γ</mi></mrow></semantics></math></inline-formula>: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mo>(</mo><mi>ρ</mi><mo>,</mo><mi>π</mi><mo>)</mo></mrow></semantics></math></inline-formula> = 0, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>σ</mi><mo><</mo><mi>β</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>T</mi><mn>1</mn></msub><mo>≤</mo><mi>γ</mi><mo>≤</mo><msub><mi>T</mi><mn>2</mn></msub></mrow></semantics></math></inline-formula>}, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>σ</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>T</mi><mn>1</mn></msub><mo><</mo><msub><mi>T</mi><mn>2</mn></msub></mrow></semantics></math></inline-formula>. We first establish an upper bound for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>N</mi><mi>π</mi></msub><mrow><mo>(</mo><mi>σ</mi><mo>,</mo><mi>T</mi><mo>,</mo><mn>2</mn><mi>T</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula> is close to 1. Then we restrict the imaginary part <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula> into a narrow strip <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mi>T</mi><mo>,</mo><mi>T</mi><mo>+</mo><msup><mi>T</mi><mi>α</mi></msup><mo>]</mo></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo>≤</mo><mn>1</mn></mrow></semantics></math></inline-formula> and prove some new zero-density results on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>N</mi><mi>π</mi></msub><mrow><mo>(</mo><mi>σ</mi><mo>,</mo><mi>T</mi><mo>,</mo><mi>T</mi><mo>+</mo><msup><mi>T</mi><mi>α</mi></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula> under specific conditions, which improves previous results when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula> near <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfrac><mn>3</mn><mn>4</mn></mfrac></semantics></math></inline-formula> and 1, respectively. The proofs rely on the zero detecting method and the Halász-Montgomery method.