Abstract
In this paper we show that Riemann's function <i>(xi)</i>, involving the Riemann’s (<i>zeta)</i> 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 <i>xi</i> function we deduce a relation between each zero of the function <i>xi</i> 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.
Highlights
Riemann's Hypothesis ([1, 3, 6]), formulated in 1859, concerns the location of the zeros of Riemann's Zeta function
The zeta function satisfies the functional equation was established by Riemann in 1859([4, 6])
Construct an infinite product model which is convergent and whose zeros are the zeros of the zeta function
Summary
Riemann's Hypothesis ([1, 3, 6]), formulated in 1859, concerns the location of the zeros of Riemann's Zeta function. Riemann presented a paper to the Berlin Academy of Mathematic In this paper, he proposed that this function, called Riemannzeta function takes values 0 on the complex plane when s=0.5+it. He proposed that this function, called Riemannzeta function takes values 0 on the complex plane when s=0.5+it This hypothesis has great significance for the world of mathematics and physics ([4]). Over a billion zeros of the function have been calculated by computers and shown that all are on this line s = 0.5+it The demonstration of this conjecture would improve in particulary the knowledge of the distribution of prime numbers. In a previous article [18] we considered studying the integral expression of the function zeta by establishing necessary conditions for the existence of zeros on the critical line.
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