Dirichlet Polynomials and Entropy

A Dirichlet polynomial d in one variable [Formula: see text] is a function of the form [Formula: see text] for some [Formula: see text]. We will show how to think of a Dirichlet polynomial as a set-theoretic bundle, and thus as an empirical distribution. We can then consider the Shannon entropy [Formula: see text] of the corresponding probability distribution, and we define its length (or, classically, its perplexity) by [Formula: see text]. On the other hand, we will define a rig homomorphism [Formula: see text] from the rig of Dirichlet polynomials to the so-called rectangle rig, whose underlying set is [Formula: see text] and whose additive structure involves the weighted geometric mean; we write [Formula: see text] , and call the two components area and width (respectively). The main result of this paper is the following: the rectangle-area formula [Formula: see text] holds for any Dirichlet polynomial d. In other words, the entropy of an empirical distribution can be calculated entirely in terms of the homomorphism h applied to its corresponding Dirichlet polynomial. We also show that similar results hold for the cross entropy.