A Generalized Discrete Bohr–Jessen-Type Theorem for the Epstein Zeta-Function
Suppose that <i>Q</i> is a positive defined <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></semantics><...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2023-02-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/11/4/799 |
| _version_ | 1797619523186065408 |
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| author | Antanas Laurinčikas Renata Macaitienė |
| author_facet | Antanas Laurinčikas Renata Macaitienė |
| author_sort | Antanas Laurinčikas |
| collection | DOAJ |
| description | Suppose that <i>Q</i> is a positive defined <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></semantics></math></inline-formula> matrix, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Q</mi><mrow><mo>[</mo><munder><mi>x</mi><mo>̲</mo></munder><mo>]</mo></mrow><mo>=</mo><msup><munder><mi>x</mi><mo>̲</mo></munder><mi mathvariant="normal">T</mi></msup><mi>Q</mi><munder><mi>x</mi><mo>̲</mo></munder></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><munder><mi>x</mi><mo>̲</mo></munder><mo>∈</mo><msup><mi mathvariant="double-struck">Z</mi><mi>n</mi></msup></mrow></semantics></math></inline-formula>. The Epstein zeta-function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ζ</mi><mo>(</mo><mi>s</mi><mo>;</mo><mi>Q</mi><mo>)</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>=</mo><mi>σ</mi><mo>+</mo><mi>i</mi><mi>t</mi></mrow></semantics></math></inline-formula>, is defined, for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>σ</mi><mo>></mo><mfrac><mi>n</mi><mn>2</mn></mfrac></mrow></semantics></math></inline-formula>, by the series <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ζ</mi><mrow><mo>(</mo><mi>s</mi><mo>;</mo><mi>Q</mi><mo>)</mo></mrow><mo>=</mo><msub><mo>∑</mo><mrow><munder><mi>x</mi><mo>̲</mo></munder><mo>∈</mo><msup><mi mathvariant="double-struck">Z</mi><mi>n</mi></msup><mo>∖</mo><mrow><mo>{</mo><munder><mn>0</mn><mo>̲</mo></munder><mo>}</mo></mrow></mrow></msub><msup><mrow><mo>(</mo><mi>Q</mi><mrow><mo>[</mo><munder><mi>x</mi><mo>̲</mo></munder><mo>]</mo></mrow><mo>)</mo></mrow><mrow><mo>−</mo><mi>s</mi></mrow></msup><mo>,</mo></mrow></semantics></math></inline-formula> and it has a meromorphic continuation to the whole complex plane. Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>⩾</mo><mn>4</mn></mrow></semantics></math></inline-formula> be even, while <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>φ</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> is an increasing differentiable function with a continuous monotonic bounded derivative <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>φ</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>φ</mi><mrow><mo>(</mo><mn>2</mn><mi>t</mi><mo>)</mo></mrow><msup><mrow><mo>(</mo><msup><mi>φ</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>≪</mo><mi>t</mi></mrow></semantics></math></inline-formula>, and the sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><mi>a</mi><mi>φ</mi><mo>(</mo><mi>k</mi><mo>)</mo><mo>}</mo></mrow></semantics></math></inline-formula> is uniformly distributed modulo 1. In the paper, it is obtained that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfrac><mn>1</mn><mi>N</mi></mfrac><mo>#</mo><mfenced separators="" open="{" close="}"><mi>N</mi><mo>⩽</mo><mi>k</mi><mo>⩽</mo><mn>2</mn><mi>N</mi><mo>:</mo><mi>ζ</mi><mo>(</mo><mi>σ</mi><mo>+</mo><mi>i</mi><mi>φ</mi><mo>(</mo><mi>k</mi><mo>)</mo><mo>;</mo><mi>Q</mi><mo>)</mo><mo>∈</mo><mi>A</mi></mfenced></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>∈</mo><mi mathvariant="script">B</mi><mo>(</mo><mi mathvariant="double-struck">C</mi><mo>)</mo></mrow></semantics></math></inline-formula>, for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>σ</mi><mo>></mo><mfrac><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mn>2</mn></mfrac></mrow></semantics></math></inline-formula>, converges weakly to an explicitly given probability measure on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi mathvariant="double-struck">C</mi><mo>,</mo><mi mathvariant="script">B</mi><mo>(</mo><mi mathvariant="double-struck">C</mi><mo>)</mo><mo>)</mo></mrow></semantics></math></inline-formula> as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>→</mo><mo>∞</mo></mrow></semantics></math></inline-formula>. |
| first_indexed | 2024-03-11T08:28:13Z |
| format | Article |
| id | doaj.art-466323e6831a47bd80580429202dfb54 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T08:28:13Z |
| publishDate | 2023-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-466323e6831a47bd80580429202dfb542023-11-16T21:54:13ZengMDPI AGMathematics2227-73902023-02-0111479910.3390/math11040799A Generalized Discrete Bohr–Jessen-Type Theorem for the Epstein Zeta-FunctionAntanas Laurinčikas0Renata Macaitienė1Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Naugarduko Str. 24, LT-03225 Vilnius, LithuaniaInstitute of Regional Development, Šiauliai Academy, Vilnius University, Vytauto Str. 84, LT-76352 Šiauliai, LithuaniaSuppose that <i>Q</i> is a positive defined <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></semantics></math></inline-formula> matrix, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Q</mi><mrow><mo>[</mo><munder><mi>x</mi><mo>̲</mo></munder><mo>]</mo></mrow><mo>=</mo><msup><munder><mi>x</mi><mo>̲</mo></munder><mi mathvariant="normal">T</mi></msup><mi>Q</mi><munder><mi>x</mi><mo>̲</mo></munder></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><munder><mi>x</mi><mo>̲</mo></munder><mo>∈</mo><msup><mi mathvariant="double-struck">Z</mi><mi>n</mi></msup></mrow></semantics></math></inline-formula>. The Epstein zeta-function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ζ</mi><mo>(</mo><mi>s</mi><mo>;</mo><mi>Q</mi><mo>)</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>=</mo><mi>σ</mi><mo>+</mo><mi>i</mi><mi>t</mi></mrow></semantics></math></inline-formula>, is defined, for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>σ</mi><mo>></mo><mfrac><mi>n</mi><mn>2</mn></mfrac></mrow></semantics></math></inline-formula>, by the series <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ζ</mi><mrow><mo>(</mo><mi>s</mi><mo>;</mo><mi>Q</mi><mo>)</mo></mrow><mo>=</mo><msub><mo>∑</mo><mrow><munder><mi>x</mi><mo>̲</mo></munder><mo>∈</mo><msup><mi mathvariant="double-struck">Z</mi><mi>n</mi></msup><mo>∖</mo><mrow><mo>{</mo><munder><mn>0</mn><mo>̲</mo></munder><mo>}</mo></mrow></mrow></msub><msup><mrow><mo>(</mo><mi>Q</mi><mrow><mo>[</mo><munder><mi>x</mi><mo>̲</mo></munder><mo>]</mo></mrow><mo>)</mo></mrow><mrow><mo>−</mo><mi>s</mi></mrow></msup><mo>,</mo></mrow></semantics></math></inline-formula> and it has a meromorphic continuation to the whole complex plane. Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>⩾</mo><mn>4</mn></mrow></semantics></math></inline-formula> be even, while <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>φ</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> is an increasing differentiable function with a continuous monotonic bounded derivative <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>φ</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>φ</mi><mrow><mo>(</mo><mn>2</mn><mi>t</mi><mo>)</mo></mrow><msup><mrow><mo>(</mo><msup><mi>φ</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>≪</mo><mi>t</mi></mrow></semantics></math></inline-formula>, and the sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><mi>a</mi><mi>φ</mi><mo>(</mo><mi>k</mi><mo>)</mo><mo>}</mo></mrow></semantics></math></inline-formula> is uniformly distributed modulo 1. In the paper, it is obtained that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfrac><mn>1</mn><mi>N</mi></mfrac><mo>#</mo><mfenced separators="" open="{" close="}"><mi>N</mi><mo>⩽</mo><mi>k</mi><mo>⩽</mo><mn>2</mn><mi>N</mi><mo>:</mo><mi>ζ</mi><mo>(</mo><mi>σ</mi><mo>+</mo><mi>i</mi><mi>φ</mi><mo>(</mo><mi>k</mi><mo>)</mo><mo>;</mo><mi>Q</mi><mo>)</mo><mo>∈</mo><mi>A</mi></mfenced></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>∈</mo><mi mathvariant="script">B</mi><mo>(</mo><mi mathvariant="double-struck">C</mi><mo>)</mo></mrow></semantics></math></inline-formula>, for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>σ</mi><mo>></mo><mfrac><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mn>2</mn></mfrac></mrow></semantics></math></inline-formula>, converges weakly to an explicitly given probability measure on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi mathvariant="double-struck">C</mi><mo>,</mo><mi mathvariant="script">B</mi><mo>(</mo><mi mathvariant="double-struck">C</mi><mo>)</mo><mo>)</mo></mrow></semantics></math></inline-formula> as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>→</mo><mo>∞</mo></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2227-7390/11/4/799Epstein zeta-functionlimit theoremweak convergenceHaar measure |
| spellingShingle | Antanas Laurinčikas Renata Macaitienė A Generalized Discrete Bohr–Jessen-Type Theorem for the Epstein Zeta-Function Mathematics Epstein zeta-function limit theorem weak convergence Haar measure |
| title | A Generalized Discrete Bohr–Jessen-Type Theorem for the Epstein Zeta-Function |
| title_full | A Generalized Discrete Bohr–Jessen-Type Theorem for the Epstein Zeta-Function |
| title_fullStr | A Generalized Discrete Bohr–Jessen-Type Theorem for the Epstein Zeta-Function |
| title_full_unstemmed | A Generalized Discrete Bohr–Jessen-Type Theorem for the Epstein Zeta-Function |
| title_short | A Generalized Discrete Bohr–Jessen-Type Theorem for the Epstein Zeta-Function |
| title_sort | generalized discrete bohr jessen type theorem for the epstein zeta function |
| topic | Epstein zeta-function limit theorem weak convergence Haar measure |
| url | https://www.mdpi.com/2227-7390/11/4/799 |
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