Closed-form calculation of infinite products of Glaisher-type related to Dirichlet series

Abstract

Building on recent work involving the computation of generalizations of Glaisher-type products over the primes by differentiation of the Euler product identity, in the present paper we generalize this approach in order to obtain closed-form expressions of more general infinite products which correspond to Dirichlet series. In this way, we obtain an elegant method to compute a variety of interesting infinite products, and some infinite double products. The Bendersky–Adamchik constants enter into a number of our results, and appear quite fundamental to these infinite products. A number of concrete examples are given in order to illustrate the general principle, including cases where these powers involve the divisor function or the Möbius function. We also consider general families of infinite products over the prime numbers (rather than the natural numbers) in order to obtain other new infinite product identities. Infinite products over terms directly involving Bendersky–Adamchik constants are considered, and these are helpful for later extending our approach to infinite double products over both the lattice of natural numbers and the lattice of prime numbers.