On Value Distribution for the Mellin Transform of the Fourth Power of the Riemann Zeta Function
In this paper, the asymptotic behavior of the modified Mellin transform <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">Z</mi><mn>2</mn&g...
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2025-01-01
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| Online Access: | https://www.mdpi.com/2075-1680/14/1/34 |
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| author | Virginija Garbaliauskienė Audronė Rimkevičienė Mindaugas Stoncelis Darius Šiaučiūnas |
| author_facet | Virginija Garbaliauskienė Audronė Rimkevičienė Mindaugas Stoncelis Darius Šiaučiūnas |
| author_sort | Virginija Garbaliauskienė |
| collection | DOAJ |
| description | In this paper, the asymptotic behavior of the modified Mellin transform <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">Z</mi><mn>2</mn></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></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>, of the fourth power of the Riemann zeta function is characterized by weak convergence of probability measures in the space of analytic functions. The main results are devoted to probability measures defined by generalized shifts <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">Z</mi><mn>2</mn></msub><mrow><mo>(</mo><mi>s</mi><mo>+</mo><mi>i</mi><mi>φ</mi><mrow><mo>(</mo><mi>τ</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> with a real increasing to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>+</mo><mo>∞</mo></mrow></semantics></math></inline-formula> differentiable functions connected to the growth of the second moment of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">Z</mi><mn>2</mn></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. It is proven that the mass of the limit measure is concentrated at the point expressed as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>h</mi><mo>(</mo><mi>s</mi><mo>)</mo><mo>≡</mo><mn>0</mn></mrow></semantics></math></inline-formula>. This is used for approximation of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>h</mi><mo>(</mo><mi>s</mi><mo>)</mo></mrow></semantics></math></inline-formula> by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">Z</mi><mn>2</mn></msub><mrow><mo>(</mo><mi>s</mi><mo>+</mo><mi>i</mi><mi>φ</mi><mrow><mo>(</mo><mi>τ</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. |
| format | Article |
| id | doaj-art-0c7228f8112d44a1b6a9d57f21adc82a |
| institution | Kabale University |
| issn | 2075-1680 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-0c7228f8112d44a1b6a9d57f21adc82a2025-01-24T13:22:13ZengMDPI AGAxioms2075-16802025-01-011413410.3390/axioms14010034On Value Distribution for the Mellin Transform of the Fourth Power of the Riemann Zeta FunctionVirginija Garbaliauskienė0Audronė Rimkevičienė1Mindaugas Stoncelis2Darius Šiaučiūnas3Institute of Regional Development, Šiauliai Academy, Vilnius University, P. Višinskio Str. 25, LT-76351 Šiauliai, LithuaniaFaculty of Business and Technologies, Šiaulių Valstybinė Kolegija, Aušros Av. 40, LT-76241 Šiauliai, LithuaniaInstitute of Regional Development, Šiauliai Academy, Vilnius University, P. Višinskio Str. 25, LT-76351 Šiauliai, LithuaniaInstitute of Regional Development, Šiauliai Academy, Vilnius University, P. Višinskio Str. 25, LT-76351 Šiauliai, LithuaniaIn this paper, the asymptotic behavior of the modified Mellin transform <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">Z</mi><mn>2</mn></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></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>, of the fourth power of the Riemann zeta function is characterized by weak convergence of probability measures in the space of analytic functions. The main results are devoted to probability measures defined by generalized shifts <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">Z</mi><mn>2</mn></msub><mrow><mo>(</mo><mi>s</mi><mo>+</mo><mi>i</mi><mi>φ</mi><mrow><mo>(</mo><mi>τ</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> with a real increasing to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>+</mo><mo>∞</mo></mrow></semantics></math></inline-formula> differentiable functions connected to the growth of the second moment of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">Z</mi><mn>2</mn></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. It is proven that the mass of the limit measure is concentrated at the point expressed as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>h</mi><mo>(</mo><mi>s</mi><mo>)</mo><mo>≡</mo><mn>0</mn></mrow></semantics></math></inline-formula>. This is used for approximation of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>h</mi><mo>(</mo><mi>s</mi><mo>)</mo></mrow></semantics></math></inline-formula> by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">Z</mi><mn>2</mn></msub><mrow><mo>(</mo><mi>s</mi><mo>+</mo><mi>i</mi><mi>φ</mi><mrow><mo>(</mo><mi>τ</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2075-1680/14/1/34limit theoremMellin transformRiemann zeta functionspace of analytic functionsweak convergence |
| spellingShingle | Virginija Garbaliauskienė Audronė Rimkevičienė Mindaugas Stoncelis Darius Šiaučiūnas On Value Distribution for the Mellin Transform of the Fourth Power of the Riemann Zeta Function Axioms limit theorem Mellin transform Riemann zeta function space of analytic functions weak convergence |
| title | On Value Distribution for the Mellin Transform of the Fourth Power of the Riemann Zeta Function |
| title_full | On Value Distribution for the Mellin Transform of the Fourth Power of the Riemann Zeta Function |
| title_fullStr | On Value Distribution for the Mellin Transform of the Fourth Power of the Riemann Zeta Function |
| title_full_unstemmed | On Value Distribution for the Mellin Transform of the Fourth Power of the Riemann Zeta Function |
| title_short | On Value Distribution for the Mellin Transform of the Fourth Power of the Riemann Zeta Function |
| title_sort | on value distribution for the mellin transform of the fourth power of the riemann zeta function |
| topic | limit theorem Mellin transform Riemann zeta function space of analytic functions weak convergence |
| url | https://www.mdpi.com/2075-1680/14/1/34 |
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