Hardy-Littlewood Constants Embedded into Infinite Products over All Positive Integers

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.

Four references added in version 2

References in corpus (1)

• Series of Reciprocal Powers of k-almost Primes

Cited by in corpus (3)

• Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli
• Survey of Dirichlet Series of Multiplicative Arithmetic Functions
• Errata and Addenda to Mathematical Constants