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 $Ξ_c(s)$ of order 1, with all zeros provably on $Re(s) = 1/2$, satisfying a functional equation $Ξ_c(s) = Ξ_c(1-s)$, whose normalized form $Z_c(s) = Ξ_c(s)/[\tfrac{1}{2}s(s-1)π^{-s/2}Γ(s/2)]$ is uniformly bounded for $Re(s) > 1 + δ$ 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 $σ= 1$ characteristic of $ζ$. The transition is controlled by a Dirichlet series $D(s) = \sum e^{-ikθ} p_k^{-s}$ whose absolute convergence for $σ> 1$ and divergence at $σ= 1$ 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.

💡 Research Summary

The paper presents a concrete construction of an entire function Ξ_c(s) of order 1 that simultaneously exhibits two hallmark features of the Riemann zeta function: (i) all non‑trivial zeros lie on the critical line Re s = ½, and (ii) a sharp bounded/unbounded transition occurs at the line σ = 1. The author begins by fixing a phase θ ∈ (0, 2π) and a small positive constant c. Using the sequence of primes p_k, a twisted Dirichlet series

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