Thinned Wallis-type prime products in residue classes modulo $2^m$

TL;DR

This paper investigates products of specific prime-related factors within residue classes modulo powers of two, establishing criteria for limits, asymptotic behavior, and expressing constants via number-theoretic functions.

Contribution

It introduces a criterion for the existence of finite limits of prime product sequences and connects these limits to Mertens-type constants and Dirichlet L-values.

Findings

Established a simple criterion for finite nonzero limits.

Proved a logarithmic asymptotic in the general case.

Expressed the limiting constant in terms of Mertens-type constants and Dirichlet L-values.

Abstract

For odd primes $p$ we consider the factors \[ A(p)=\frac{p-\chi_4(p)}{p+\chi_4(p)}, \qquad \chi_4(p)= \begin{cases} 1,&p\equiv 1\pmod 4, \\ -1,&p\equiv 3\pmod 4, \end{cases} \] 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, and express the limiting constant in terms of Mertens-type constants in arithmetic progressions (hence in terms of Dirichlet $L$-values).