A Hybrid Computational Framework for Riemann Hypothesis Analysis: Integrating Multiple Mathematical Methods with Geometric Visualization

This paper presents a computational framework that integrates multiple math-ematical methods for analyzing the Riemann hypothesis through numerical com-putation. The framework combines gap-based prime discovery algorithms, consecutivesquare-root approximation techniques, and geometric vector field visualization meth-ods to investigate zero distributions and prime counting functions. Through sys-tematic computational analysis of ranges up to t ≤ 100, the paper demonstratesconsistent numerical patterns supporting existing theoretical predictions. This im-plementation achieves high computational accuracy in zero detection (29/29 the-oretical zeros found) and prime distribution analysis (median normalized error of0.0026). While these results provide computational evidence rather than math-ematical proof, they offer new tools for numerical investigation of the Riemannhypothesis and demonstrate the utility of multi-method integration in computa-tional number theory.