The Riemann Hypothesis Remains Unresolved: Fundamental Discrepancy in Computational Methods for the Widely Accepted Nontrivial Zeros of the Riemann Zeta Function, ζ(s)

The long-standing mathematical problem, the Riemann Hypothesis, states that all nontrivial zeros of the Riemann zeta function ζ(s) lie on the line Re(s) =1/2. This study examines the computational methods used to determine nontrivial zeros, such as ½ + 14.13472514i, ½ + 21.02203964i, ½ + 25.01085758i, and others of ζ(s).The analysis finds that these methods are derived from the equation ξ(s) =(s/2) (s-1) (π)-s/2 Γ(s/2) ζ(s), by assuming that zeros of functions ζ(s) and ξ(s) are identical. However, the graphical examination of the locations of zeros of both functions, suggests this assumption is incorrect, rendering the computational methods flawed. Consequently, the widely accepted zeros computed using these methods, might not actually be zeros of ζ(s). In addition, the formula for the number of zeros N(T) on the line s=1/2 in a specific interval [0, T], based on the same assumption is also invalid. The study offers a new perspective on the Riemann Hypothesis by highlighting potential flaws in existing methods used for computing zeros of ζ(s). This finding could contribute to ongoing efforts to resolve the hypothesis.