An investigation of the non-trivial zeros of the Riemann zeta function

Abstract

The zeros of the Riemann zeta function outside the critical strip are the so-called 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 we have $\zeta(s) = \zeta(1-\bar{s})$. 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 sides of the critical line within the critical strip $\Re(s) \in \, ]0,1[$, to say the Riemann hypothesis is true.Comment: 16 page