Using System of equations to investigate the Riemann Hypothesis
Résumé
The Riemann Hypothesis (RH), one of the most profound and longstanding open problems in mathematics, posits that all nontrivial zeros of the Riemann zeta function $\zeta(s)$ lie on the critical line $\Re(s)=0.5$. In this paper, we present a proof of RH by employing a reductio ad absurdum approach. Assuming the existence of zeros off the critical line but within the critical strip, we derive a contradiction through the rigorous construction of structured, consistent, and well-posed systems of equations using the Abel-Plana formula. This transforms RH from a complex analytical problem into a computational framework by analyzing the numerical solutions to these systems. Our findings are consistent with the classical results of Hardy and Littlewood regarding zeros in the critical strip and confirm that all nontrivial zeros lie on the critical line. This approach provides a systematic framework for addressing RH and sheds new light on the interplay between symmetry, structured systems of equations, and the zeros of the zeta function.
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