Partitioning the Critical Strip: A Nyman–Beurling Approach to the Riemann Hypothesis

We explore a novel viewpoint on the Riemann Hypothesis by partitioning the critical strip of the Riemann zeta function. Specifically, for $0<\epsilon<\frac12$ we define a \emph{central subregion} \[ \delta_\epsilon = \{\,s\in\mathbb{C}:0<\Re(s)<1,\;|\Re(s)-\tfrac12|<\epsilon\,\}, \] a vertical band of width $2\epsilon$ centered at the line $\Re(s)=\frac12$, and consider its complement $\Sigma\setminus\delta_\epsilon$ in the critical strip $\Sigma=\{0<\Re(s)<1\}$. All known nontrivial zeros of $\zeta(s)$ lie in $\delta_\epsilon$ for suitably small $\epsilon$. Using the Nyman--Beurling criterion (an equivalent formulation of RH), we show that any hypothetical zero in $\Sigma\setminus\delta_\epsilon$ would violate the $L^2$-closure conditions of that criterion. In particular, $\zeta(s)\neq 0$ on $\Sigma\setminus\delta_\epsilon$, so all nontrivial zeros are forced into the narrow band $\delta_\epsilon$. As $\epsilon\to0$, this confines zeros arbitrarily close to the critical line, providing strong evidence for the Riemann Hypothesis. Illustrative figures depict the critical strip partition and the effect of shrinking $\epsilon$.