Inequalities in the complex domain and the Riemann hypothesis

Abstract

The sign ζ(z) of the function at the points of the critical band is considered. The representation of the ordering in the domain of complex numbers z1<z2 <=>Re(z1) )<Re(z2) and at Re(z1)=Re(z2) with Im(z1)<Im(z2). Since the area where ζ(z) <0 accommodates the subdomain where Re(z)<0, the allocation of the latter can be carried out on a PC. The points of intersection of the boundaries of the regions Re(z)<0 and Im(z)<0 and are the roots of ζ(z). It is impossible to distinguish the real part of the root from ½, since the root z1 <1/2 corresponds to the root z2 = 1-z1 in the same band. But it will lie in the band where ζ(z) >0, and not equal to zero. If z1 >½, then 1- z1 will lie in the region where ζ(z) <0, which is impossible if ζ(z2) =0. Therefore, z2= z1 = ½.