Riemann Zeros as Attractors of a Smooth Gradient Flow

We present a dynamical reformulation of the Riemann Hypothesis (RH) via the smooth potential ((s) = (-|(s)|^2)) defined on the critical strip. The zeros of ((s)) are global maxima of (). We study the gradient flow (Ze flow) of () and formulate two main conjectures. Conjecture 1 states that for any fixed (t), the function ((+it)) attains its unique maximum at (/2). We prove that Conjecture 1 is conditionally equivalent to RH, assuming the Lindelöf Hypothesis (LH). Conjecture 2 states that for each fixed (t), the Ze flow has exactly one attracting fixed point in ((0,1)), located at (/2). We prove that Conjecture 2 is equivalent to RH for simple zeros. The reverse direction (RH ⇒ Conjecture 2) is verified numerically for (t ^4); a rigorous unconditional proof requires establishing (2/2(1/2,t) < 0) for all (t), which we state as an open problem (Conjecture 2’). Numerical experiments for 1000 values of (t) in ((0,10^4]) confirm the unique maximum and attractor at (/2). A heuristic connection between the Fourier transform of ((1/2+it)) and the explicit formula is presented as an open problem. This work does not prove RH; it provides a novel dynamical framework that may inspire new analytical or computational approaches.