A Verified-Height Region of Horizontal Monotonicity for the Riemann ξ-Function
Year: 2026
Type: preprint
Abstract: This preprint proves a verified-height region of horizontal monotonicity for the Riemann ξ-function. Let H = 3,000,175,332,800 be the height up to which Platt and Trudgian rigorously verified that all nontrivial zeros lie on the critical line. We prove that Re(ξ′/ξ)(1/2 + δ + it) > 0 for 0 < δ ≤ 1/2 and |t| ≤ H − sqrt(1/4 − δ²). In particular, horizontal monotonicity holds throughout the full rectangle 1/2 < σ ≤ 1, |t| ≤ H − 1/2. The proof groups zeros using the functional-equation symmetry and derives an exact sign formula for the contribution of a hypothetical off-critical pair. Such a pair can contribute negatively only inside a disk whose radius is its horizontal displacement from the critical line. Since every nontrivial zero satisfies |Re(ρ) − 1/2| < 1/2, zeros above the verified height cannot contribute negatively in the stated region. This removes the unknown-tail estimates used in the previous version and yields a stronger result with a substantially shorter proof. No assumption is made about the location or multiplicity of zeros above H, and no new computation of zeta zeros is used.
Author: Paim Kessler, Felix Cristiano
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7203927575
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