On the Applications of the Generalized Littlewood Theorem Concerning Integrals of the Logarithm of Analytical Functions to Elliptic Functions
Recently, we have established and used the generalized Littlewood theorem concerning contour integrals of the logarithm of analytical function to obtain new criteria equivalent to the Riemann hypothesis. Later, the same theorem was applied to calculate certain infinite sums and study the properties...
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| Format: | Article |
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MDPI AG
2023-06-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/12/6/595 |
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| author | Sergey Sekatskii |
| author_facet | Sergey Sekatskii |
| author_sort | Sergey Sekatskii |
| collection | DOAJ |
| description | Recently, we have established and used the generalized Littlewood theorem concerning contour integrals of the logarithm of analytical function to obtain new criteria equivalent to the Riemann hypothesis. Later, the same theorem was applied to calculate certain infinite sums and study the properties of zeroes of a few analytical functions. In this study, we apply this approach to elliptic functions of Jacobi and Weierstrass. Numerous sums over inverse powers of zeroes and poles are calculated, including some results for the Jacobi elliptic functions <i>sn</i>(<i>z</i>, <i>k</i>) and others understood as functions of the index <i>k</i>. The consideration of the case of the derivative of the Weierstrass rho-function, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>℘</mo><mi>z</mi></msub><mo stretchy="false">(</mo><mi>z</mi><msub><mo>,</mo><mrow></mrow></msub><mi>τ</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>, leads to quite easy and transparent proof of numerous equalities between the sums over inverse powers of the lattice points <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>+</mo><mi>n</mi><mi>τ</mi></mrow></semantics></math></inline-formula> and “demi-lattice” points <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>+</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>+</mo><mi>n</mi><mi>τ</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>+</mo><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo stretchy="false">)</mo><mi>τ</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>+</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>+</mo><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo stretchy="false">)</mo><mi>τ</mi></mrow></semantics></math></inline-formula>. We also prove theorems showing that, in most cases, fundamental parallelograms contain exactly one simple zero for the first derivative <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>θ</mi><mn>1</mn></msub><mo>′</mo><mo stretchy="false">(</mo><mi>z</mi><mo>|</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> of the elliptic theta-function and the Weierstrass <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ζ</mi></semantics></math></inline-formula>-function, and that far from the origin of coordinates such zeroes of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ζ</mi></semantics></math></inline-formula>-function tend to the positions of the simple poles of this function. |
| first_indexed | 2024-03-11T02:46:39Z |
| format | Article |
| id | doaj.art-46540ff4dcfe49c5a0017b4e75de6ba1 |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-11T02:46:39Z |
| publishDate | 2023-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-46540ff4dcfe49c5a0017b4e75de6ba12023-11-18T09:17:14ZengMDPI AGAxioms2075-16802023-06-0112659510.3390/axioms12060595On the Applications of the Generalized Littlewood Theorem Concerning Integrals of the Logarithm of Analytical Functions to Elliptic FunctionsSergey Sekatskii0Laboratory of Biological Electron Microscopy, Institute of Physics, BSP 419, Ecole Polytechnique Fédérale de Lausanne and Department of Fundamental Biology, Faculty of Biology and Medicine, University of Lausanne, CH1015 Lausanne, SwitzerlandRecently, we have established and used the generalized Littlewood theorem concerning contour integrals of the logarithm of analytical function to obtain new criteria equivalent to the Riemann hypothesis. Later, the same theorem was applied to calculate certain infinite sums and study the properties of zeroes of a few analytical functions. In this study, we apply this approach to elliptic functions of Jacobi and Weierstrass. Numerous sums over inverse powers of zeroes and poles are calculated, including some results for the Jacobi elliptic functions <i>sn</i>(<i>z</i>, <i>k</i>) and others understood as functions of the index <i>k</i>. The consideration of the case of the derivative of the Weierstrass rho-function, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>℘</mo><mi>z</mi></msub><mo stretchy="false">(</mo><mi>z</mi><msub><mo>,</mo><mrow></mrow></msub><mi>τ</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>, leads to quite easy and transparent proof of numerous equalities between the sums over inverse powers of the lattice points <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>+</mo><mi>n</mi><mi>τ</mi></mrow></semantics></math></inline-formula> and “demi-lattice” points <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>+</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>+</mo><mi>n</mi><mi>τ</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>+</mo><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo stretchy="false">)</mo><mi>τ</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>+</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>+</mo><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo stretchy="false">)</mo><mi>τ</mi></mrow></semantics></math></inline-formula>. We also prove theorems showing that, in most cases, fundamental parallelograms contain exactly one simple zero for the first derivative <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>θ</mi><mn>1</mn></msub><mo>′</mo><mo stretchy="false">(</mo><mi>z</mi><mo>|</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> of the elliptic theta-function and the Weierstrass <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ζ</mi></semantics></math></inline-formula>-function, and that far from the origin of coordinates such zeroes of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ζ</mi></semantics></math></inline-formula>-function tend to the positions of the simple poles of this function.https://www.mdpi.com/2075-1680/12/6/595generalized Littlewood theoremlogarithm of an analytical functionelliptic functionszeroes and poles of analytical functioninfinite sums |
| spellingShingle | Sergey Sekatskii On the Applications of the Generalized Littlewood Theorem Concerning Integrals of the Logarithm of Analytical Functions to Elliptic Functions Axioms generalized Littlewood theorem logarithm of an analytical function elliptic functions zeroes and poles of analytical function infinite sums |
| title | On the Applications of the Generalized Littlewood Theorem Concerning Integrals of the Logarithm of Analytical Functions to Elliptic Functions |
| title_full | On the Applications of the Generalized Littlewood Theorem Concerning Integrals of the Logarithm of Analytical Functions to Elliptic Functions |
| title_fullStr | On the Applications of the Generalized Littlewood Theorem Concerning Integrals of the Logarithm of Analytical Functions to Elliptic Functions |
| title_full_unstemmed | On the Applications of the Generalized Littlewood Theorem Concerning Integrals of the Logarithm of Analytical Functions to Elliptic Functions |
| title_short | On the Applications of the Generalized Littlewood Theorem Concerning Integrals of the Logarithm of Analytical Functions to Elliptic Functions |
| title_sort | on the applications of the generalized littlewood theorem concerning integrals of the logarithm of analytical functions to elliptic functions |
| topic | generalized Littlewood theorem logarithm of an analytical function elliptic functions zeroes and poles of analytical function infinite sums |
| url | https://www.mdpi.com/2075-1680/12/6/595 |
| work_keys_str_mv | AT sergeysekatskii ontheapplicationsofthegeneralizedlittlewoodtheoremconcerningintegralsofthelogarithmofanalyticalfunctionstoellipticfunctions |