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

Ralph Furmaniak

TL;DR

The paper constructs an explicit order-$1$ entire function $Ξ_c(s)$ with all zeros on the critical line $Re(s)=1/2$ and a symmetry $Ξ_c(s)=Ξ_c(1-s)$, together with a normalized ratio $Z_c(s)$ that remains bounded for $Re(s)>1$ yet blows up on the line $Re(s)=1$. This is achieved via a Hadamard product with carefully perturbed zeros, guided by a Fourier-analytic mechanism that links the zero perturbations to a damped Dirichlet-series component, and a dyadic large-sieve argument that yields unconditional control. The authors further show that the ratio $Z_c/Z_0$ is itself a Dirichlet series with a generalized Euler product, bounded in $Re(s)>1$ and unbounded on $Re(s)=1$, with all zeros on the critical line; its zeros/poles reflect the perturbed and unperturbed spectra. A spectral identity connects the perturbations to a main Fourier sum, and a Phragmén–Lindelöf transfer explains the sharp bounded/unbounded transition. Numerically, the zeros exhibit highly regular spacing and value distributions that are isotropic but constrained, highlighting a distinct contrast with the microscopic GUE statistics of the Riemann zeta function. The construction demonstrates that an RH-like zero distribution can coexist with a sigma-1 transition, providing a concrete, analyzable object that straddles essential features of $ζ$ without its arithmetic complexity.

Abstract

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 construction uses Hadamard products with controlled zero perturbations; the transition is proved unconditionally via a dyadic large-sieve argument and a Phragmén-Lindelöf transfer.

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

TL;DR

The paper constructs an explicit order- entire function with all zeros on the critical line and a symmetry , together with a normalized ratio that remains bounded for yet blows up on the line . This is achieved via a Hadamard product with carefully perturbed zeros, guided by a Fourier-analytic mechanism that links the zero perturbations to a damped Dirichlet-series component, and a dyadic large-sieve argument that yields unconditional control. The authors further show that the ratio is itself a Dirichlet series with a generalized Euler product, bounded in and unbounded on , with all zeros on the critical line; its zeros/poles reflect the perturbed and unperturbed spectra. A spectral identity connects the perturbations to a main Fourier sum, and a Phragmén–Lindelöf transfer explains the sharp bounded/unbounded transition. Numerically, the zeros exhibit highly regular spacing and value distributions that are isotropic but constrained, highlighting a distinct contrast with the microscopic GUE statistics of the Riemann zeta function. The construction demonstrates that an RH-like zero distribution can coexist with a sigma-1 transition, providing a concrete, analyzable object that straddles essential features of without its arithmetic complexity.

Abstract

We construct a single explicit entire function of order 1, with all zeros provably on , satisfying a functional equation , whose normalized form is uniformly bounded for yet satisfies . The function thus satisfies an analogue of the Riemann Hypothesis together with the sharp bounded/unbounded transition at characteristic of . The construction uses Hadamard products with controlled zero perturbations; the transition is proved unconditionally via a dyadic large-sieve argument and a Phragmén-Lindelöf transfer.
Paper Structure (17 sections, 7 theorems, 19 equations, 2 figures)

This paper contains 17 sections, 7 theorems, 19 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Method
  3. The Explicit Construction
  4. Parameters
  5. The perturbed zeros
  6. The Spectral Identity
  7. The Fourier-Analytic Mechanism
  8. Proof of the Main Theorem
  9. The Ratio $Z_c/Z_0$ and Dirichlet Series
  10. Almost-Periodicity
  11. Observations
  12. First zeros of $Z_c$
  13. Comparison with $\zeta$ zero statistics
  14. Value distribution
  15. Numerical evidence for the linearization condition
  16. ...and 2 more sections

Key Result

Theorem 1.1

For any $c > 0$, define $\Xi_c(s)$ as in §sec:construction, with $Z_c(s) = \Xi_c(s)/[\tfrac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)]$. Then: Parts (a)--(c) hold by construction; (d) via the spectral identity (Proposition prop:spectral) with linearization error controlled by Lemma lem:lin; and (e) via a Phragmén--Lindelöf argument (§sec:proof). Moreover, the ratio $Z_c/Z_0$ is itself a Dirichlet series w

Figures (2)

  • Figure 1: Complex-plane image of $\log f(\sigma+it)$ for $t\in[20,300]$. Left: $\log(Z_c/Z_0)$; center: $\log\zeta$; right: $\log L(s,\chi_4)$. Rows: $\sigma=1.25, 1.0, 0.75$. Color encodes $t$ (yellow$=$early, purple$=$late). Note the ${\sim}20{\times}$ difference in scale between the left column and the others.
  • Figure 2: Shape comparison: $\log(Z_c/Z_0)$ rescaled to match the RMS of $\log\zeta$ (left) vs. $\log\zeta$ (right). The rescaled $Z_c$ fills a banded region (reflecting the coherent trigonometric structure) rather than the isotropic spirals of $\log\zeta$ (reflecting multiplicative independence of primes).

Theorems & Definitions (17)

  • Theorem 1.1: Main Theorem
  • Lemma 1.2: Linearization convergence
  • proof
  • Lemma 2.1: Convergence
  • proof
  • Lemma 2.2: Well-definedness of the Hadamard product
  • proof
  • Remark 2.3
  • Proposition 3.1: Spectral Identity
  • proof
  • ...and 7 more