Work

Two problems in comparative prime number theory: the five-way race modulo 11, and the sign of the integrated prime counting error

Year: 2026

Type: preprint

Abstract: We resolve two problems of Hamieh, Kadiri, Martin and Ng in comparative prime number theory. Under GRH and the usual linear-independence hypothesis, we compute all 120 logarithmic densities in the five-way prime number race among the quadratic residues modulo 11, prove that they form eight strictly ordered symmetry classes, and give rigorous enclosures. We also classify the residue-class pairs for which the integrated prime counting error is eventually negative. Both parts use Turing-certified zero lists and rigorous bounds for omitted zeros.

Source: Zenodo (CERN European Organization for Nuclear Research)

DOI: 10.5281/zenodo.22208025

Authors: Liu, Ziyang, Wang, Yunxuan, Shi, Yao

Institutions: National University of Singapore, ETH Zurich, The Hong Kong University of Science and Technology (Guangzhou)

Language: English

Topic: Analytic Number Theory Research

Subfield: Algebra and Number Theory

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

OpenAlex ID: W7204799174

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