Bounded Solutions of a Complex Differential Equation for the Riemann Hypothesis

In this paper, we consider the representation of the Riemann zeta function ζ defined by Abel's summation formula. Using the differential equations, we show that |(1-s)ζ(s) -s̄ζ(1-s̄)| ≠ 0 for any point s in the critical strip except the critical line. This result suggests an asymmetry of (1-s)ζ(s) across the critical strip. This does not contradict the Riemann functional equation but prove that non-trivial zeros cannot lie off the critical line. These results are consistent with the Riemann Hypothesis and suggest that non-trivial zeros lie on the critical line.