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. 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. If this theory is proven to be correct, it means we will be able to know the sequential order of the prime numbers. This function is useful in number theory for investigating the anomalous behavior of prime numbers. 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. The Riemann zeta function plays a momentous part while analyzing the number theory and has applications in applied statistics, probability theory and Physics. 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. Moreover, it was well known to him that all non-trivial zeros are exhibiting symmetry with respect to the critical line. (commonly known as trivial zeros) has an infinite number of zeros in the critical strip of complex numbers between the lines = 0 and = 1. Euler-Riemann found that the function equals zero for all negative even integers: −2, −4, −6. The Riemann zeta function is valid for all complex number, for the line = 1. Jamal Salah 1 ,*, Hameed Ur Rehman 2, Iman Al- Buwaiqi 2ġ Department of Basic Science, College of Health and Applied Sciences, A’ Sharqiyah University, OmanĢ Department of Mathematics, Center for Language and Foundation Studies, A’ Sharqiyah University, Oman The Non-Trivial Zeros of The Riemann Zeta Function through Taylor Series Expansion and Incomplete Gamma Function
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