Some Remarks on a Product Expansion

Abstract

The product expansion given by \(\prod\limits_{n = 2}^\infty {\left( {1 - {n^{ - 3}}} \right)} = 1 + \sum\limits_{n = 2}^\infty {e*} \left( n \right){n^{ - 3}},\operatorname{Re} \;s > 1,\)defines e*(n) as a product partition function. Namely, e*(n) represents the excess of the number of representations of n > 1 as a product of an even number of distinct factors (also called parts in the sequel) > 1 over those into an odd number of such parts, the order of the parts in the product being ignored. To our knowledge, this function has not been considered in the literature so far. In this preliminary study, we obtain in Sections 1 and 2 a basic property of this function. Let f(k) denote the value of e*(p 1…p k ), where p 1,…, P k are distinct arbitrarily chosen primes. We show that as \(k \to \infty ,\;\log \mid f\left( k \right)\mid /k\)is unbounded, and in fact \(\mathop {\lim \;\sup }\limits_{k \to \infty } \;\frac{{\log \mid f\left( k \right)\mid }}{{k\;\log \;k}} = 1.\)More generally, a similar result holds for a certain weighted partition function associated with e*(n). In Section 3 we make several remarks on the expansion of infinite products associated with certain additive and multiplicative partition functions. Some open problems are mentioned at the end of this note.