The Admissible Domain of the Non-Trivial Zeros of the Riemann Zeta Function

The zeros of the Riemann zeta function outside the critical strip are the trivial zeros. While many zeros of the Riemann zeta function are located on the critical line $\Re(s)=1/2$, the non-existence of zeros in the remaining part of the critical strip $\Re(s) \in \, ]0,1[$ remains to be proven. The Riemann zeta functional leads to a relationship between the zeros of the Riemann zeta function on either sides of the critical line. Given $s$ a complex number and $\bar{s}$ its complex conjugate, if $s$ is a zero of the Riemann zeta function in the critical strip $\Re(s) \in \, ]0,1[$, then $1-\bar{s}$ is also a zero. As the Riemann hypothesis states that all non-trivial zeros lie on the critical line $\Re(s) = 1/2$, it is enough to show there are no zeros on either of the intervals $\Re(s) \in \, ]0, \frac{1}{2}[$ or $]\frac{1}{2},1[$, to say the Riemann hypothesis is true.