# Dirichlet Polynomials and Entropy

@article{Spivak2021DirichletPA, title={Dirichlet Polynomials and Entropy}, author={David I. Spivak and Timothy Hosgood}, journal={Entropy}, year={2021}, volume={23} }

A Dirichlet polynomial d in one variable y is a function of the form d(y)=anny+⋯+a22y+a11y+a00y for some n,a0,…,an∈N. 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 H(d) of the corresponding probability distribution, and we define its length (or, classically, its perplexity) by L(d)=2H(d). On the other hand, we will define a rig homomorphism h:Dir→Rect from the rig of Dirichlet…

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### Polynomial functors and Shannon entropy

- MathematicsArXiv
- 2022

Past work shows that one can associate a notion of Shannon entropy to a Dirichlet polynomial, regarded as an empirical distribution. Indeed, entropy can be extracted from any 3 ∈ Dir by a two-step…

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