Thinned Wallis-Type Prime Products in Residue Classes Modulo \(2^m\)

2026/02/28 by Mike Winkler
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Rings, Modules, and Algebras #math.GM #msc:11M06 #msc:11M20 #msc:11N13 #msc:11N37

paper · pdf · doi:10.5281/zenodo.21678054

published as INTEGERS 26 (2026) #A90 · 11 pages, 4 tables

openalex publication_date 2026/07/29 · arxiv created 2026/07/30 · openalex created_date 2026/07/30 · openalex updated_date 2026/07/30 · arxiv updated 2026/07/31

Abstract

For odd primes p we consider the factors A(p)=(p-χ₄(p))/(p+χ₄(p)), where χ₄ is the quadratic Dirichlet character modulo 4, and study products of A(p) restricted to unions of residue classes modulo 2^m. We give a simple criterion for the existence of a finite nonzero limit, prove a logarithmic asymptotic in the general case, express the limiting constant in terms of Mertens-type constants in arithmetic progressions and Dirichlet L-values, and give reproducible high-precision computations of the resulting constants.

Citations

• Euler constants from primes in arithmetic progression

Related

• Products in Residue Classes
• Residue of a Mod 5 Euler Product
• On the existence of products of primes in arithmetic progressions
• Product of an integer free of small prime factors and prime in arithmetic progression
• Distribution of residues modulo p using the Dirichlet's class number formula