An Analytical and Numerical Detour for the Riemann Hypothesis
From the functional equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mo>=<...
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| Formato: | article |
| Lenguaje: | EN |
| Publicado: |
MDPI AG
2021
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| Materias: | |
| Acceso en línea: | https://doaj.org/article/f0985cf4521640868e0216dc31f15937 |
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| Sumario: | From the functional equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mo>=</mo><mi>F</mi><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mi>s</mi></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of Riemann’s zeta function, this article gives new insight into Hadamard’s product formula. The function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>S</mi><mn>1</mn></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mo>=</mo><mi>d</mi><mrow><mo>(</mo><mrow><mi>ln</mi><mi>F</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>/</mo><mi>d</mi><mi>s</mi></mrow></semantics></math></inline-formula> and its family of associated <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>S</mi><mi>m</mi></msub></mrow></semantics></math></inline-formula> functions, expressed as a sum of rational fractions, are interpreted as meromorphic functions whose poles are the poles and zeros of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>F</mi></semantics></math></inline-formula> function. This family is a mathematical and numerical tool which makes it possible to estimate the value <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of the function at a point <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>=</mo><mi>x</mi><mo>+</mo><mi>i</mi><mi>y</mi><mo>=</mo><mover accent="true"><mi>x</mi><mo>˙</mo></mover><mo>+</mo><mo>½</mo><mo>+</mo><mi>i</mi><mi>y</mi></mrow></semantics></math></inline-formula> in the critical strip <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">S</mi></semantics></math></inline-formula> from a point <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>𝓈</mo><mo>=</mo><mo>½</mo><mo>+</mo><mi>i</mi><mi>y</mi></mrow></semantics></math></inline-formula> on the critical line <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>ℒ</mo></semantics></math></inline-formula>.Generating estimates <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>S</mi><mi>m</mi><mo>∗</mo></msubsup><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>S</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> at a given point requires a large number of adjacent zeros, due to the slow convergence of the series. The process allows a numerical approach of the Riemann hypothesis (RH). The method can be extended to other meromorphic functions, in the neighborhood of isolated zeros, inspired by the Weierstraß canonical form. A final and brief comparison is made with the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ζ</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi></mrow></semantics></math></inline-formula> functions over finite fields. |
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