An Explicit Entire Function of Order One with All Zeros on a Line and Bounded in a Half-Plane

We construct a single explicit entire function ${\mathrm{\Xi }}_c(s)$ of order 1, with all zeros provably on $Re(s)=1\mathrm{/}2$, satisfying a functional equation ${\mathrm{\Xi }}_c(s)={\mathrm{\Xi }}_c(1-s)$, whose normalized form $Z_c(s)={\mathrm{\Xi }}_c(s)\mathrm{/}[\frac{1}{2}s(s-1){\pi }^{-s\mathrm{/}2}\mathrm{\Gamma }(s\mathrm{/}2)]$ is uniformly bounded for $Re(s)>1+\delta$ yet satisfies ${\sup }_t\mathrm{\mid }{Z}_c(1+it)\mathrm{\mid }=+\mathrm{\infty }$. The function thus satisfies an analogue of the Riemann Hypothesis together with the sharp bounded/unbounded transition at $\sigma =1$ characteristic of $\zeta$. The transition is controlled by a Dirichlet series $D(s)=\sum e^{-ik\theta }{p}_k^{-s}$ whose absolute convergence for $\sigma >1$ and divergence at $\sigma =1$ drive the dichotomy. The key technical input is a dyadic large-sieve estimate establishing the linearization condition that connects the Hadamard product to $D$. The construction and proofs were developed in collaboration with Claude (Anthropic); see Acknowledgments.