Analyzing Riemann's hypothesis

Mercedes Orús–Lacort, Román Orús and Christophe Jouis*

In this paper we perform a detailed analysis of Riemann's hypothesis, dealing with the zeros of the analytically-extended zeta function. We use the functional equation  for complex numbers s such that 0<Re(s)<1, and the reduction to the absurd method, where we use an analytical study based on a complex function and its modulus as a real function of two real variables, in combination with a deep numerical analysis, to show that the real part of the non-trivial zeros of the Riemann zeta function is equal to ½, to the best of our resources. This is done in two steps. First, we show what would happen if we assumed that the real part of s has a value between 0 and 1 but different from 1/2, arriving at a possible contradiction for the zeros. Second, assuming that there is no real value y such that ζ(1/2+yi)=0, by applying the rules of logic to negate a quantifier and the corresponding Morgan's law we also arrive at a plausible contradiction. Finally, we analyze what conditions should be satisfied by y∈ℝ such that ζ(1/2+yi)=0. While these results are valid to the best of our numerical calculations, we do not observe and foresee any tendency for a change. Our findings open the way towards assessing the validity of Riemman's hypothesis from a fresh and new mathematical perspective.

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Published on: Jun 16, 2023 Pages: 75-82

Full Text PDF Full Text HTML DOI: 10.17352/amp.000083
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