[Paper Review] Hardy-Littlewood Constants Embedded into Infinite Products over All Positive Integers

Richard J. Mathar|arXiv (Cornell University)|Mar 13, 2009

Analytic Number Theory Research10 references3 citations

TL;DR

This paper re-expresses Hardy-Littlewood constants as infinite products over k-almost primes, leveraging k-almost prime zeta functions to enable numerical evaluation through series expansions. It derives novel product identities and transformations, embedding classical constants into a unified framework based on arithmetic multiplicative functions and special zeta functions.

ABSTRACT

A group of infinite products over low-order rational polynomials evaluated at the sequence of prime numbers is loosely called the Hardy-Littlewood constants. In this manuscript we look at them as factors embedded in a super-product over primes, semiprimes, 3-almost primes etc. Numerical tables are derived by transformation into series over k-almost prime zeta functions. Alternative product representations in a basis of k-almost prime products associated with Euler's formula for the Riemann zeta function are also pointed out.

Motivation & Objective

• To reinterpret classical Hardy-Littlewood constants as infinite products over k-almost primes, rather than over all integers.
• To enable numerical computation of these constants by transforming products into series over k-almost prime zeta functions.
• To derive alternative product representations using Euler-type formulas and special functions like the Gamma and Riemann zeta functions.
• To establish a systematic framework for expressing rational and transcendental constants arising from rational polynomials evaluated at integers.
• To explore the algebraic and analytic structure of infinite products by classifying terms via the total number of prime factors with multiplicity (Ω(n)).

Proposed method

• Classifies integers by Ω(n), the total number of prime factors with multiplicity, to reorganize infinite products over all n ≥ 2 into products over k-almost primes.
• Employs the k-almost prime zeta function P_k(s) = ∑_{Ω(n)=k} n^{-s} as a core analytical tool for series-based evaluation.
• Applies logarithmic expansion and series inversion techniques, including the use of the Möbius function, to relate products to zeta functions.
• Derives identities by manipulating Euler’s product formula for ζ(s) and its sign-switched variants to express products in terms of ζ_k(s) and auxiliary functions A_k^{(s)}, Q_k^{(s)}, F_k^{(s)}.
• Uses functional equations and reflection formulas of the Gamma function to simplify infinite product expressions into closed forms.
• Introduces and analyzes auxiliary functions (A_k^{(s)}, Q_k^{(s)}, F_k^{(s)}) to generate new product identities with rational and transcendental values.

Experimental results

Research questions

• RQ1 How can Hardy-Littlewood-type constants be re-expressed as products over k-almost primes instead of over all integers?
• RQ2 What is the role of the k-almost prime zeta function P_k(s) in enabling numerical evaluation of such infinite products?
• RQ3 Can closed-form identities be derived for infinite products of rational functions over integers by classifying terms via Ω(n)?
• RQ4 What algebraic structures emerge when products over n ≥ 2 are decomposed by the number of prime factors with multiplicity?
• RQ5 How do auxiliary functions like A_k^{(s)}, Q_k^{(s)}, and F_k^{(s)} unify and generalize known product identities involving zeta and Gamma functions?

Key findings

• The product ∏_{n=2}^∞ (n²−1)/(n²+1) is decomposed into ∏_{k=1}^∞ ∏_{Ω(n)=k} (n²−1)/(n²+1), with numerical values tabulated for k=1 to 4 and s=2 to 8.
• For s even and k=1, the product over primes yields rational numbers: ∏_p (p^s−1)/(p^s+1) = ζ(2s)/ζ²(s) = 2B_{2s}/(C(2s)B_s²), with Bernoulli numbers B_n.
• The product over all n≥2 of (1−1/n^s) equals 1/∏_{k=1}^∞ ζ_k(s), where ζ_k(s) is the k-almost prime zeta function of the second kind.
• Specific identities are derived, such as ∏_{Ω(n)=k} (1−1/n^{s+1}) = 1/(ζ_k(s+1) A_k^{(s)}), linking zeta functions to auxiliary functions.
• Numerical values for k=1, s=2 yield ∏_p (p²−1)/(p²+1) = 0.4 = 2/5, and for s=4, k=1, the product is 6/7 ≈ 0.857142857.
• The paper derives 50+ identities involving A_k^{(s)}, Q_k^{(s)}, F_k^{(s)}, showing that rational and transcendental constants emerge from structured infinite products over arithmetic classes.