Examination of the Proof of Riemann's hypothesis and the conditions for it to hold for Generalized {\zeta}-functions. It holds for Riemann's {\zeta}-function and the resulting distribution of primes

Abstract

The proof of Riemann's hypothesis follows from the simple logic, that when two properties are related, i.e. these equations are zero i.e. {\zeta}(z) = {\zeta}(1-z) = 0 while they have the proven 1-1 property of Riemann's function {\zeta}(z), then the Hypothesis will be true because it has been proved from before simultaneously that Riemann's functional equations hold. Therefore, there will be apply {\zeta}(z) = {because {\zeta}(z) = {\zeta}(1-z) = 0 and also because {\zeta}(z) as well as {\zeta}(1-z) are 1-1}. This, being true, will give the direction of all non-trivial roots to all be on the critical line, with a value of 1/2 on the real axis. Furthermore, to strengthen the proof we consider important requirements that clearly prove that 100% of the roots of {\zeta}(z)=0 are only on the critical line. There is of course the case {\zeta}(qz)=0 with q>1/2 in R, which has different critical lines, but all those who deal with the Hypothesis of Riemann, do not mention it, because they claim that this form was not defined by Riemann from the beginning. But if we generalize the hypothesis to {\zeta}(qz)=0,q>1/2 in R then we have infinite critical lines that originate from and are related to the baseline critical line of Re(z)=1/2,but are not identical to it and lie within the interval (0,1). In summary and by analytically examining and other similar functions to {\zeta}(z), we get zeros with Re(z) in the interval (0,1) for the cases {\zeta}(qz)=0 where q>1/2 in R that mentioned, for Generalized case G-{\zeta}(z,q)=0 with at least 2 general cases out of 5 which is the generalized sum including {\zeta}(z) and end for the Davenport Heilbronn case, which has zeros of a periodic Dirichlet series with different critical lines, within the interval (0,1). We are likely to have other constructed cases with the relevance of {\zeta}(z).