A Framework for the Zeta–Eta Ratio and Zero Distribution in the Critical Strip

This preprint presents a novel ratio-based approach to the Riemann Hypothesis by comparing the functional equations of the Riemann zeta function ζ(s) and the Dirichlet eta function η(s). Defining  

g(s)= (1–2¹⁻ˢ)/(1–2ˢ) and  

k(s)= 2ˢ πˢ⁻¹ sin(πs/2) Γ(1–s),  

we study H(s)=g(s)/k(s) and Φ(s)=H(s)–1. We prove that H(s) is analytic throughout the open critical strip 0<Re(s)<1 and that Φ(s) has exactly one real zero at s=½. We outline a contour‐integration argument via the argument principle suggesting a single zero in the full strip, and show how the Identity Theorem would force H(s)≡1 once an infinite, accumulating set of zeros is established. Key gaps—rigorous phase‐bounding on all contour sides, zero‐accumulation for Φ, and complete analytic continuation across branch cuts—are highlighted. This draft serves as a clear roadmap for turning these preliminary results into a full proof that all nontrivial zeros of ζ(s) lie on Re(s)=½.