# Dirichlet Polynomials and Entropy

^{*}

## Abstract

**:**

## 1. Introduction

- Section 2: We recall the definitions of Dirichlet polynomials and set-theoretic bundles, along with their rig structures, from [1] (one important thing to note is the following: Dirichlet polynomials are well-studied objects in the setting of complex analysis, but we cannot apply tools from this area to our setting, because we have only natural number coefficients and non-negative exponents); we then study the equivalence between these two notions.
- Section 3: We explain how empirical probability distributions correspond to set-theoretic bundles (and thus to Dirichlet polynomials).
- Section 4: We define the rig homomorphism $h:\mathsf{Dir}\to \mathsf{Rect}$ that we wish to study, whose codomain is a rig encoding the weighted geometric mean; we prove some useful computational results and give some explicit examples.
- Section 5: We define the entropy $H\left(d\right)$ of a Dirichlet polynomial using the classical notion of Shannon entropy; we give some explicit examples; we prove the main result of this paper (Theorem 1), relating entropy to the rig homomorphism defined in the previous section.
- Section 6: We try to provide some intuition for the image $h\left(d\right)$ of a Dirichlet polynomial under the rig homomorphism, in terms of coding schemes.

#### Prerequisites

## 2. Dirichlet Polynomials and Bundles

**Definition**

**1.**

**Example**

**1.**

**Definition**

**2.**

**Example**

**2.**

**Example**

**3.**

**Lemma**

**1.**

**Proof.**

**Definition**

**3.**

**Corollary**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Definition**

**4.**

## 3. Bundles as Empirical Distributions

**Example**

**4.**

**Remark**

**1.**

**Remark**

**2.**

## 4. Area and Width

**Definition**

**5.**

**Proposition**

**1.**

**Proof.**

**Definition**

**6.**

**Lemma**

**3.**

- 1.
- $h\left(a\right)=(a,1)$;
- 2.
- $A(a\xb7d)=aA\left(d\right)$;
- 3.
- $W(a\xb7d)=W\left(d\right)$.

**Proof.**

**Corollary**

**2.**

**Proof.**

**Corollary**

**3.**

**Proof.**

**Example**

**5.**

**Example**

**6.**

**Example**

**7.**

## 5. Length

**Remark**

**3.**

**Definition**

**7.**

**Example**

**8.**

**Example**

**9.**

**Example**

**10.**

**Theorem**

**1.**

**Proof.**

## 6. Interpreting Area, Length, and Width

**arbitrary**morphisms of bundles. (If, however, we restrict to only morphisms given by pushforward, then [7] tells us (via Faddeev’s theorem) that the only possible functorial definition of entropy is given by the relative entropy, i.e., the difference of the entropies of the source and the target). This makes it seem rather bad to work with a category (such as that of bundles) instead of simply a rig (such as that of Dirichlet polynomials). Of course, this isn’t an entirely satisfactory answer, since we do care about the notion of morphisms for Dirichlet polynomials (for example, Corollary 2 tells us that the width can be expressed in terms of the number of certain morphisms). In light of Theorem 1, however, we might consider the following possibility: both area and length can be expressed in terms of $d\left(0\right)$, $d\left(1\right)$, and $d\left[i\right]$ (for $i\in d\left(0\right)$), and we could define the width by $W\left(d\right)A\left(d\right)/L\left(d\right)$.

## 7. Cross Entropy

**Definition**

**8.**

**Remark**

**4.**

**Remark**

**5.**

- 1.
- we recover the “uncrossed” notions when we take $d=e$, and
- 2.
- Theorem 2 holds.

**Theorem**

**2.**

**Proof.**

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Spivak, D.I.; Myers, D.J. Dirichlet Polynomials form a Topos. arXiv
**2020**, arXiv:2003.04827. [Google Scholar] - Leinster, T. Basic Category Theory; Cambridge University Press: Cambridge, UK, 2014. [Google Scholar]
- Fritz, T.; Perrone, P. A probability monad as the colimit of spaces of finite samples. Theory Appl. Categ.
**2019**, 34, 170–220. [Google Scholar] - Baez, J.C.; Hoffnung, A.E.; Walker, C.D. Higher-Dimensional Algebra VII: Groupoidification. Theory Appl. Categ.
**2010**, 24, 489–553. [Google Scholar] - Shannon, C.E. A Mathematical Theory of Communication. Bell Syst. Tech. J.
**1948**, 27, 379–423. [Google Scholar] [CrossRef] [Green Version] - KidzSearch Wiki. Rectangle Facts for Kids. Available online: https://wiki.kidzsearch.com/wiki/Rectangle (accessed on 18 August 2021).
- Baez, J.C.; Fritz, T.; Leinster, T. A Characterization of Entropy in Terms of Information Loss. Entropy
**2011**, 13, 1945–1957. [Google Scholar] [CrossRef] - Huffman, D.A. A method for the construction of minimum-redundancy codes. Proc. IRE
**1952**, 40, 1098–1101. [Google Scholar] [CrossRef] - Baez, J.C.; Fritz, T. A Bayesian characterization of relative entropy. Theory Appl. Categ.
**2014**, 29, 421–456. [Google Scholar]

**Figure 1.**Our convention for naming the sides of a rectangle, from [6].

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Spivak, D.I.; Hosgood, T.
Dirichlet Polynomials and Entropy. *Entropy* **2021**, *23*, 1085.
https://doi.org/10.3390/e23081085

**AMA Style**

Spivak DI, Hosgood T.
Dirichlet Polynomials and Entropy. *Entropy*. 2021; 23(8):1085.
https://doi.org/10.3390/e23081085

**Chicago/Turabian Style**

Spivak, David I., and Timothy Hosgood.
2021. "Dirichlet Polynomials and Entropy" *Entropy* 23, no. 8: 1085.
https://doi.org/10.3390/e23081085