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A Speedy New Proof of the Riemann's Hypothesis

Received: 20 March 2021     Accepted: 24 April 2021     Published: 14 May 2021
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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.

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), 2021. Published by Science Publishing Group

Keywords

Riemann Hypothesis, Weierstrass, Zeta Function, Holomorphic Function, Infinite Product, Dirichlet Series

References
[1] J. Hadamard, «On the zeros distribution of the zeta function and its aritmetic consequences», Bulletin of the S. M. F, Tome 24, 1896 – Ecole centrale- Paris, France.
[2] C.J. de La Vallée Poussin, « On the zeros of the Riemann function», C.R. Acad. Sciences Paris, 163 (1916), 418-421, 1916 – Louvain University – Louvain, Belgium
[3] H.M. Edwards, «Reimann's Zeta Function», Academic Press, New York, London, 1974 – Harvard University - USA
[4] S.J. Patterson, «An introduction to the theory of the Riemann zeta function», Ed. 1995 – University of Gottingen - Germany
[5] K. Knopp, «Weierstrass’s factor-Theorem», Theory of Functions, part II, New York, 1996– Eberhard Karl University – Germany.
[6] G. Lachaud, « Riemann Hypothesis », La Recherche, 2001- Institut of Mathématiques, Marseille – CNRS - France
[7] S. Vento, «Factorization of holomophic function, Hardy Spaces» 2003, Study and research works, D. Laboratoire Analyse, Géométrie et Applications - Université Paris 1 – France.
[8] V. Garcia-Morales 2007, «On the nontrivial zeros of the Dirichlet eta function» - Dept. de Fisica de la Terra- University of Valencia - Spain
[9] H. Kobayashi 2016, « Local extrema of the E(t) function and The Riemann Hypothesis » – Princeton University – N-J – USA
[10] F. Stenger 2018, «A new proof of Newman’s Conjecture and generalization» – UCLA Mathematics Department – L-A – USA
[11] G. Mussardo and A. LeClair, 2018, «Generalized Riemann Hypothesis and stochastic time series» J. of Statistical Mechanics: Theorie and experiment, Vol 2018 – June 2018
[12] M. Atiyah, 2018, «The Riemann Hypothesis»- Edinburgh University – Edinburgh, G-B
[13] A. Connes 2019, «An essay on the Riemann Hypothesis» – College de France – Paris, France
[14] C. Chuanmio, 2020, «Local geometric proof of Riemann Hypothesis» – Hunan Normal University -,Hunan, China
[15] A. Dobner 2020, «A new proof of Newman’s Conjecture and generalization» – UCLA Mathematics Department – L-A – USA
[16] L. Tao and W. Juhao, 2020, «A proof of Riemann Hypothesis» - School of Science, Southwest University of Science and Technology, Mianyang, Sichuan, China and Stanford University, Stanford, California, USA
[17] F. Alhargan, 2021, «A concise Proof of the Riemann Hypothesis based on Hadamard Product» - PhD, Eng. Senior Researcher at PSDSARC – Riyadh, Saudi Arabia
[18] J-M. Coranson-Beaudu, «Proof of the Riemann hypothesis» 2020 - African Journal of Mathematics and Computer Science Research, AJMCSR – Martinique, France.
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    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

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    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

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    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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  • @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}
    }
    

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    AU  - Jean-Max Coranson-Beaudu
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    PY  - 2021
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    DO  - 10.11648/j.pamj.20211002.13
    T2  - Pure and Applied Mathematics Journal
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    JO  - Pure and Applied Mathematics Journal
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    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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Author Information
  • Free Researcher Department of Fluid Mechanics, Sorbonne University/Paris 6, Martinique, France

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