The Non-Trivial Zeros of The Riemann Zeta Function through Taylor Series Expansion and Incomplete Gamma Function

Abstract

The Riemann zeta (ζ) function ζ(s) = ∞ n=1 1 n s is valid for all complex number s = x + iy : Re(s) > 1, for the line x = 1.Euler-Riemann found that the function equals zero for all negative even integers: -2, -4, -6, • • • (commonly known as trivial zeros) has an infinite number of zeros in the critical strip of complex numbers between the lines x = 0 and x = 1.Moreover, it was well known to him that all non-trivial zeros are exhibiting symmetry with respect to the critical line x = 1 2 .As a result, Riemann conjectured that all of the non-trivial zeros are on the critical line, this hypothesis is known as the Riemann hypothesis.The Riemann zeta function plays a momentous part while analyzing the number theory and has applications in applied statistics, probability theory and Physics.The Riemann zeta function is closely related to one of the most challenging unsolved problems in mathematics (the Riemann hypothesis) which has been classified as the 8th of Hilbert's 23 problems.This function is useful in number theory for investigating the anomalous behavior of prime numbers.If this theory is proven to be correct, it means we will be able to know the sequential order of the prime numbers.Numerous approaches have been applied towards the solution of this problem, which includes both numerical and geometrical approaches, also the Taylor series of the Riemann zeta function, and the asymptotic properties of its coefficients.Despite the fact that there are around 10 13 , non-trivial zeros on the critical line, we cannot assume that the Riemann Hypothesis (RH) is necessarily true unless a lucid proof is provided.Indeed, there are differing viewpoints not only on the Riemann Hypothesis's reliability, but also on certain basic conclusions see for example [16] in which the author justifies the location of non-trivial zero subject to the simultaneous occurrence of ζ(s) = ζ(1 -s) = 0, and omitting the impact of an indeterminate form ∞.0, that appears in Riemann's approach.In this study we also consider the simultaneous occurrence ζ(s) = ζ(1 -s) = 0 but we adopt an element-wise approach of the Taylor series by expanding n -x for all n = 1, 2, 3, • • • at the real parts of the non-trivial zeta zeros lying in the critical strip for s = α + iy is a non-trivial zero of ζ(s), we first expand each term n -x at α then at 1 -α.Then In this sequel, we evoke the simultaneous occurrence of the non-trivial zeta function zeros ζ(s) = ζ(1 -s) = 0, on the critical strip by the means of different representations of Zeta function.Consequently, proves that Riemann Hypothesis is likely to be true.

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