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

Abstract

We construct a single explicit entire function ${\Xi }_c(s)$ of order 1, with all zeros provably on $Re(s)=1/2$, satisfying a functional equation ${\Xi }_c(s)={\Xi }_c(1-s)$, whose normalized form $Z_c(s)={\Xi }_c(s)/[\frac{1}{2}s(s-1){\pi }^{-s/2}\Gamma (s/2)]$ is uniformly bounded for $Re(s)>1+\delta$ yet satisfies ${\sup }_t|Z_c(1+it)|=+\infty$. 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.