Explicit zeros of the Davenport–Heilbronn function with real part greater than one
Year: 2026
Type: preprint
Abstract: Eight zeros of the Davenport–Heilbronn function f(s) = Σ (1, κ, −κ, −1, 0)[n mod 5] n−s with real part greater than one are proved by Rouché's theorem, with f enclosed in ball arithmetic through an exact smoothed functional equation (no prime sums, no constants from the literature), at heights between 2.66·1012 and 7.56·1013, with real parts 1.0008–1.0051. They are located by lattice reduction on the phases of the primes ≡ 2, 3 (mod 5). The note explains why search and proof meet near height 1013–1014 and stop well short of the supremum 1.120362 of Bombieri and Ghosh. No explicit such zero of this function was found in the literature searched. Version 2 (1 November 2026). The description of version 1 said four zeros were proved; the note proves eight. Version 2 also corrects a real-part range in the paragraph on why the pipeline stops (1.0008–1.0051, not 1.002–1.005), cites Balanzario and Sánchez-Ortiz (2007) for zeros with real part above one of two sister series, and adds the eight proved zeros in machine-readable form (proved_zeros.csv). Written jointly with Claude, an AI system made by Anthropic, which carried out much of the computation, the derivations and the drafting; the human author is responsible for the content. Not peer reviewed. The package contains the PDF, its LaTeX source, the scripts and result files it cites, and the dated literature-search log behind its priority statements.
Author: Berzai, Nathan
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7224870068