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

Abstract

For odd primes $p$ we consider the factors $A(p)=(p-\ddot{I}{‡}_4(p))/(p+\ddot{I}{‡}_4(p))$, where $\ddot{I}{‡}_4$ 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.

11 pages, 4 tables