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

TL;DR

This paper constructs an explicit order-one entire function with all zeros on a line, satisfying a Riemann Hypothesis analogue and exhibiting a boundedness transition at a critical line, driven by a specific Dirichlet series.

Contribution

It introduces a new explicit entire function of order one with zeros on a line, satisfying a functional equation and boundedness properties similar to the Riemann zeta function.

Findings

Function has all zeros on the critical line Re(s)=1/2

Normalized form is bounded for Re(s)>1+δ but unbounded on the line Re(s)=1

Boundedness transition is driven by a Dirichlet series with a divergence at σ=1

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 $Z_c(s) = \Xi_c(s)/[\tfrac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)]$ 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.