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 $\zeta(s) = 2^{s}\pi^{s-1}\sin{(\displaystyle \pi s/2)}\Gamma(1-s)\zeta(1-s)$ for complex numbers $s$ such that $0<{\rm Re(s)}<1$ 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 $1/2$ to the best of our resources. This is done in two steps. Firstly, 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. Secondly assuming that there is no real value $y$ such that $\zeta\left(1/2 +yi \right)=0$ by applying the rules of logic to negate a quantifier and the corresponding Morgan's law we also arrive to a plausible contradiction. Finally, we analyze what conditions should be satisfied by $y \in \mathbb R$ such that $\zeta(\displaystyle 1/2 +yi)=0$. 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.