In this paper we show that Riemann's function (xi), involving the Riemann’s (zeta) function, is holomorphic and is expressed as a convergent infinite polynomial product in relation to their zeros and their conjugates. Our work will be done on the critical band in which non-trivial zeros exist. Our approach is to use the properties of power series and infinite product decomposition of holomorphic functions. We take inspiration from the Weierstrass method to construct an infinite product model which is convergent and whose zeros are the zeros of the zeta function. By applying the symetric functional equation of the xi function we deduce a relation between each zero of the function xi and its conjugate. Because of the convergence of the infinite product, and that the elementary polynomials of the second degree of this same product are irreducible into the complex set, then this relation is well determined. The apparent simplicity of the reasoning is based on the fundamental theorems of Hadamard and Mittag-Leffler. We obtain the sought result: the real part of all zeros is equal to ½. This article proves that the Riemann’ hypothesis is true. Our perspectives for a next article are to apply this method to Dirichlet series, as a generalization of the Riemann function.
| Published in | Pure and Applied Mathematics Journal (Volume 10, Issue 2) |
| DOI | 10.11648/j.pamj.20211002.13 |
| Page(s) | 62-67 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2024. Published by Science Publishing Group |
Riemann Hypothesis, Weierstrass, Zeta Function, Holomorphic Function, Infinite Product, Dirichlet Series
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APA Style
Jean-Max Coranson-Beaudu. (2021). A Speedy New Proof of the Riemann's Hypothesis. Pure and Applied Mathematics Journal, 10(2), 62-67. https://doi.org/10.11648/j.pamj.20211002.13
ACS Style
Jean-Max Coranson-Beaudu. A Speedy New Proof of the Riemann's Hypothesis. Pure Appl. Math. J. 2021, 10(2), 62-67. doi: 10.11648/j.pamj.20211002.13
AMA Style
Jean-Max Coranson-Beaudu. A Speedy New Proof of the Riemann's Hypothesis. Pure Appl Math J. 2021;10(2):62-67. doi: 10.11648/j.pamj.20211002.13
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Download@article{10.11648/j.pamj.20211002.13,
author = {Jean-Max Coranson-Beaudu},
title = {A Speedy New Proof of the Riemann's Hypothesis},
journal = {Pure and Applied Mathematics Journal},
volume = {10},
number = {2},
pages = {62-67},
doi = {10.11648/j.pamj.20211002.13},
url = {https://doi.org/10.11648/j.pamj.20211002.13},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20211002.13},
abstract = {In this paper we show that Riemann's function (xi), involving the Riemann’s (zeta) function, is holomorphic and is expressed as a convergent infinite polynomial product in relation to their zeros and their conjugates. Our work will be done on the critical band in which non-trivial zeros exist. Our approach is to use the properties of power series and infinite product decomposition of holomorphic functions. We take inspiration from the Weierstrass method to construct an infinite product model which is convergent and whose zeros are the zeros of the zeta function. By applying the symetric functional equation of the xi function we deduce a relation between each zero of the function xi and its conjugate. Because of the convergence of the infinite product, and that the elementary polynomials of the second degree of this same product are irreducible into the complex set, then this relation is well determined. The apparent simplicity of the reasoning is based on the fundamental theorems of Hadamard and Mittag-Leffler. We obtain the sought result: the real part of all zeros is equal to ½. This article proves that the Riemann’ hypothesis is true. Our perspectives for a next article are to apply this method to Dirichlet series, as a generalization of the Riemann function.},
year = {2021}
}
TY - JOUR T1 - A Speedy New Proof of the Riemann's Hypothesis AU - Jean-Max Coranson-Beaudu Y1 - 2021/05/14 PY - 2021 N1 - https://doi.org/10.11648/j.pamj.20211002.13 DO - 10.11648/j.pamj.20211002.13 T2 - Pure and Applied Mathematics Journal JF - Pure and Applied Mathematics Journal JO - Pure and Applied Mathematics Journal SP - 62 EP - 67 PB - Science Publishing Group SN - 2326-9812 UR - https://doi.org/10.11648/j.pamj.20211002.13 AB - In this paper we show that Riemann's function (xi), involving the Riemann’s (zeta) function, is holomorphic and is expressed as a convergent infinite polynomial product in relation to their zeros and their conjugates. Our work will be done on the critical band in which non-trivial zeros exist. Our approach is to use the properties of power series and infinite product decomposition of holomorphic functions. We take inspiration from the Weierstrass method to construct an infinite product model which is convergent and whose zeros are the zeros of the zeta function. By applying the symetric functional equation of the xi function we deduce a relation between each zero of the function xi and its conjugate. Because of the convergence of the infinite product, and that the elementary polynomials of the second degree of this same product are irreducible into the complex set, then this relation is well determined. The apparent simplicity of the reasoning is based on the fundamental theorems of Hadamard and Mittag-Leffler. We obtain the sought result: the real part of all zeros is equal to ½. This article proves that the Riemann’ hypothesis is true. Our perspectives for a next article are to apply this method to Dirichlet series, as a generalization of the Riemann function. VL - 10 IS - 2 ER -
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