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| Format: | Preprint |
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2026
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| Online-Zugang: | https://arxiv.org/abs/2601.18687 |
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| _version_ | 1866918318537965568 |
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| author | Furmaniak, Ralph |
| author_facet | Furmaniak, Ralph |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_18687 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An Explicit Entire Function of Order One with All Zeros on a Line and Bounded in a Half-Plane Furmaniak, Ralph Number Theory 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. |
| title | An Explicit Entire Function of Order One with All Zeros on a Line and Bounded in a Half-Plane |
| topic | Number Theory |
| url | https://arxiv.org/abs/2601.18687 |