Attempting to Prove the Riemann Hypothesis through the Reflection Formula

Abstract

The Riemann Hypothesis, originally proposed by the eminent mathematician Bernard Riemann in 1859, remains one of the most profound challenges in number theory. It posits that all non-trivial zeros of the Riemann zeta function {\zeta}(s) are concentrated precisely along the critical line where the real part equals 1/2. In this paper, our aim is to present an attempt to prove this conjecture. Our approach relies on the use of the reflection formula. By applying this tool with precision and insight, we can conclusively establish that {\xi}(s)^2 (Riemann's {\xi}-function) is valid only when Re(s)=1/2. As a direct consequence of this determination, we can assert that every zero of both {\xi}(s)^2 and {\xi}(s) has a real part equal to 1/2. This, in turn, leads us to the tentative conclusion that the real part of all non-trivial zeros of the zeta function is consistently 1/2.