A Unifying Principle in the Theory of Modular Relations
The Voronoĭ summation formula is known to be equivalent to the functional equation for the square of the Riemann zeta function in case the function in question is the Mellin tranform of a suitable function. There are some other famous summation formulas which are treated as independent of the modula...
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MDPI AG
2023-01-01
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| Online Access: | https://www.mdpi.com/2227-7390/11/3/535 |
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| author | Guodong Liu Kalyan Chakraborty Shigeru Kanemitsu |
| author_facet | Guodong Liu Kalyan Chakraborty Shigeru Kanemitsu |
| author_sort | Guodong Liu |
| collection | DOAJ |
| description | The Voronoĭ summation formula is known to be equivalent to the functional equation for the square of the Riemann zeta function in case the function in question is the Mellin tranform of a suitable function. There are some other famous summation formulas which are treated as independent of the modular relation. In this paper, we shall establish a far-reaching principle which furnishes the following. Given a zeta function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Z</mi><mo>(</mo><mi>s</mi><mo>)</mo></mrow></semantics></math></inline-formula> satisfying a suitable functional equation, one can generalize it to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Z</mi><mi>f</mi></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> in the form of an integral involving the Mellin transform <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>(</mo><mi>s</mi><mo>)</mo></mrow></semantics></math></inline-formula> of a certain suitable function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> and process it further as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mover accent="true"><mi>Z</mi><mo>˜</mo></mover><mi>f</mi></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. Under the condition that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>(</mo><mi>s</mi><mo>)</mo></mrow></semantics></math></inline-formula> is expressed as an integral, and the order of two integrals is interchangeable, one can obtain a closed form for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mover accent="true"><mi>Z</mi><mo>˜</mo></mover><mi>f</mi></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. Ample examples are given: the Lipschitz summation formula, Koshlyakov’s generalized Dedekind zeta function and the Plana summation formula. In the final section, we shall elucidate Hamburger’s results in light of RHBM correspondence (i.e., through Fourier–Whittaker expansion). |
| first_indexed | 2024-03-11T09:35:47Z |
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| language | English |
| last_indexed | 2024-03-11T09:35:47Z |
| publishDate | 2023-01-01 |
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| spelling | doaj.art-a9f306d3e98b4ea2bc4c96d47dde61e42023-11-16T17:20:59ZengMDPI AGMathematics2227-73902023-01-0111353510.3390/math11030535A Unifying Principle in the Theory of Modular RelationsGuodong Liu0Kalyan Chakraborty1Shigeru Kanemitsu2Department of Mathematics and Statistics, Huizhou University, Huizhou 516007, ChinaKSCSTE-Kerala School of Mathematics, Kozhikode 673571, IndiaKSCSTE-Kerala School of Mathematics, Kozhikode 673571, IndiaThe Voronoĭ summation formula is known to be equivalent to the functional equation for the square of the Riemann zeta function in case the function in question is the Mellin tranform of a suitable function. There are some other famous summation formulas which are treated as independent of the modular relation. In this paper, we shall establish a far-reaching principle which furnishes the following. Given a zeta function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Z</mi><mo>(</mo><mi>s</mi><mo>)</mo></mrow></semantics></math></inline-formula> satisfying a suitable functional equation, one can generalize it to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Z</mi><mi>f</mi></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> in the form of an integral involving the Mellin transform <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>(</mo><mi>s</mi><mo>)</mo></mrow></semantics></math></inline-formula> of a certain suitable function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> and process it further as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mover accent="true"><mi>Z</mi><mo>˜</mo></mover><mi>f</mi></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. Under the condition that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>(</mo><mi>s</mi><mo>)</mo></mrow></semantics></math></inline-formula> is expressed as an integral, and the order of two integrals is interchangeable, one can obtain a closed form for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mover accent="true"><mi>Z</mi><mo>˜</mo></mover><mi>f</mi></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. Ample examples are given: the Lipschitz summation formula, Koshlyakov’s generalized Dedekind zeta function and the Plana summation formula. In the final section, we shall elucidate Hamburger’s results in light of RHBM correspondence (i.e., through Fourier–Whittaker expansion).https://www.mdpi.com/2227-7390/11/3/535summation formulasmodular relationMellin tranformRiemann zeta functionfunctional equation |
| spellingShingle | Guodong Liu Kalyan Chakraborty Shigeru Kanemitsu A Unifying Principle in the Theory of Modular Relations Mathematics summation formulas modular relation Mellin tranform Riemann zeta function functional equation |
| title | A Unifying Principle in the Theory of Modular Relations |
| title_full | A Unifying Principle in the Theory of Modular Relations |
| title_fullStr | A Unifying Principle in the Theory of Modular Relations |
| title_full_unstemmed | A Unifying Principle in the Theory of Modular Relations |
| title_short | A Unifying Principle in the Theory of Modular Relations |
| title_sort | unifying principle in the theory of modular relations |
| topic | summation formulas modular relation Mellin tranform Riemann zeta function functional equation |
| url | https://www.mdpi.com/2227-7390/11/3/535 |
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