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

Mike Winkler

Abstract

For odd primes $p$ we consider the factors \[ A(p)=\frac{p-χ_4(p)}{p+χ_4(p)}, \qquad χ_4(p)= \begin{cases} 1,&p\equiv 1\pmod 4, \\ -1,&p\equiv 3\pmod 4, \end{cases} \] 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).

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

Abstract

For odd primes we consider the factors and study products of restricted to unions of residue classes modulo . 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 -values).
Paper Structure (8 sections, 5 theorems, 38 equations)

This paper contains 8 sections, 5 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminary facts
  3. A structural identity
  4. Asymptotic behavior and the balance criterion
  5. The limiting constant
  6. Examples
  7. Concluding remarks
  8. Numerical values

Key Result

Lemma 1

For every odd prime $p$,

Theorems & Definitions (9)

  • Lemma 1
  • proof
  • Theorem 1
  • proof : Proof of Theorem \ref{['thm:asymptotic']}
  • Corollary 1
  • Proposition 1: Williams Williams
  • Theorem 2
  • proof : Proof of Theorem \ref{['thm:constant']}
  • Remark 1