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

Abstract

The Riemann zeta <img src=image/13425982_01.gif> function <img src=image/13425982_02.gif> is valid for all complex number <img src=image/13425982_03.gif>, for the line <img src=image/13425982_04.gif> = 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 <img src=image/13425982_04.gif> = 0 and <img src=image/13425982_04.gif> = 1. Moreover, it was well known to him that all non-trivial zeros are exhibiting symmetry with respect to the critical line <img src=image/13425982_05.gif>. 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 10¹³, 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 <img src=image/13425982_06.gif>, and omitting the impact of an indeterminate form <img src=image/13425982_07.gif>, that appears in Riemann's approach. In this study, we also consider the simultaneous occurrence <img src=image/13425982_06.gif> but we adopt an element-wise approach of the Taylor series by expanding <img src=image/13425982_08.gif> for all <img src=image/13425982_09.gif> = 1, 2, 3, ... at the real parts of the non-trivial zeta zeros lying in the critical strip for <img src=image/13425982_10.gif> is a non-trivial zero of <img src=image/13425982_11.gif>, we first expand each term <img src=image/13425982_08.gif> at <img src=image/13425982_12.gif> then at <img src=image/13425982_13.gif>. Then in this sequel, we evoke the simultaneous occurrence of the non-trivial zeta function zeros<img src=image/13425982_06.gif>, on the critical strip by the means of different representations of Zeta function. Consequently, proves that Riemann Hypothesis is likely to be true.