Analyzing Riemann's hypothesis

Abstract

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 ζ(s)=2sπs−1sin(πs/2)Γ(1−s)ζ(1−s) 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.