THE PLANCHEREL LINE AND MELLIN COMPLETENESS CRITERIAFOR THE RIEMANN ZETA FUNCTION

The Nyman--Beurling criterion establishes that the Riemann Hypothesis (RH) is equivalent to a completeness condition in $L^2(\mathbb{R}_{>0}, dx)$, whose Mellin--Parseval identity holds on $\re(s) = \tfrac{1}{2}$. Beurling's parametrised theorem shows that moving to $L^p$ with $p > 1$ replaces RH with the weaker condition $\zeta(s) \neq 0$ for $\re(s) > 1/p$, approaching the Prime Number Theorem (PNT) as $p \to 1^+$. We prove a general completeness theorem for Mellin-type function systems in weighted $L^2$ spaces. Let $\sigma_0 \geq 1$ and let $\{f_\alpha\}$ be a system in $L^2(\mathbb{R}_{>0},\, x^{2\sigma_0-1}\, dx)$ whose Mellin transforms factor as $m_\alpha(s)\,\zeta(s)\, g(s)$, where $g$ is non-vanishing and rapidly decreasing on $\re(s) = \sigma_0$ and the multiplier family $\{m_\alpha\}$ is total on this line. Then the system is unconditionally complete: the only input is $\zeta(\sigma_0+it) \neq 0$, which for $\sigma_0 > 1$ follows from the Euler product and for $\sigma_0 = 1$ is the PNT. As a concrete instance, we show that dilated Jacobi theta-function differences are complete in $L^2(\mathbb{R}_{>0},\, x\,dx)$. The result identifies the Plancherel line as a structural barrier: any Mellin completeness argument operating on $\re(s) \geq 1$ cannot encode information about zeros inside the critical strip, and therefore cannot be equivalent to RH.