An evaluation to the Taylor critical strip finding method against the analytic Laurent series one

In the present paper, this author will try to revisit the proof of my Riemann Hypothesis by extending the Taylor Series Approximation to the Laurent Series which is in fact can overcome the deficiencies in the divergence problem for the classical Riemann Zeta Function for the case of 0 < s < 1. In addition, this author also introduces the multiplication factor of (1-2(1-s)) to the zeta function when combined with each other to form the Dirichlet Eta function and hence we may reuse nearly all of my previous papers’ computed results. Next this author also tries to optimize the Riemann Zeta Function’s empirical root model equation and compare with the inverse power index 2 and 3. It is shown that if we multiply a factor to the inner part of the cotangent function, then both of the empirical equations can be minimized while there may another parts for the maximization which may indeed constitute the business simplex method “primal & dual” problems. The last issue of this paper is to focus in the number theory, elliptical curves and elliptical functions together with the cryptography. In a simplified and less abstract language, there is a one-to-one correspondence between the elliptical curves and the elliptical functions. Hence, in one way, we may solve the Weierstrass P function and establish the corresponding Fourier series with a suitable period and lattice such that we may get the essential congruence modular formula for the cryptography that has been shown in the Apostol and Washington’s books. On the other hand, from the respective elliptical curves, we may also obtain an algorithmic procedures (that will be shown in the section before conclusion) for solving the integral modular on the elliptical curves and thus establish the cryptography. In brief, the present paper will extend the proof for the Riemann Hypothesis for the case of 0 < s < 1. The paper also optimizes the empirical Riemann Zeta Root model equations. It also helps us to get an application in the field of elliptical curve cryptography from the proof of truth to the Riemann Hypothesis.