| Summary: | A Dirichlet polynomial <i>d</i> in one variable <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="normal">y</mi></semantics></math></inline-formula> is a function of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mrow><mo>(</mo><mi mathvariant="normal">y</mi><mo>)</mo></mrow><mo>=</mo><msub><mi>a</mi><mi>n</mi></msub><msup><mi>n</mi><mi mathvariant="normal">y</mi></msup><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>a</mi><mn>2</mn></msub><msup><mn>2</mn><mi mathvariant="normal">y</mi></msup><mo>+</mo><msub><mi>a</mi><mn>1</mn></msub><msup><mn>1</mn><mi mathvariant="normal">y</mi></msup><mo>+</mo><msub><mi>a</mi><mn>0</mn></msub><msup><mn>0</mn><mi mathvariant="normal">y</mi></msup></mrow></semantics></math></inline-formula> for some <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>,</mo><msub><mi>a</mi><mn>0</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>a</mi><mi>n</mi></msub><mo>∈</mo><mi mathvariant="double-struck">N</mi></mrow></semantics></math></inline-formula>. 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>(</mo><mi>d</mi><mo>)</mo></mrow></semantics></math></inline-formula> of the corresponding probability distribution, and we define its <i>length</i> (or, classically, its <i>perplexity</i>) by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mrow><mo>(</mo><mi>d</mi><mo>)</mo></mrow><mo>=</mo><msup><mn>2</mn><mrow><mi>H</mi><mo>(</mo><mi>d</mi><mo>)</mo></mrow></msup></mrow></semantics></math></inline-formula>. On the other hand, we will define a rig homomorphism <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>h</mi><mo lspace="0pt">:</mo><mi mathvariant="sans-serif">Dir</mi><mo>→</mo><mi mathvariant="sans-serif">Rect</mi></mrow></semantics></math></inline-formula> from the rig of Dirichlet polynomials to the so-called <i>rectangle rig</i>, whose underlying set is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">R</mi><mrow><mo>⩾</mo><mn>0</mn></mrow></msub><mo>×</mo><msub><mi mathvariant="double-struck">R</mi><mrow><mo>⩾</mo><mn>0</mn></mrow></msub></mrow></semantics></math></inline-formula> and whose additive structure involves the weighted geometric mean; we write <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>h</mi><mo>(</mo><mi>d</mi><mo>)</mo><mo>=</mo><mo>(</mo><mi>A</mi><mo>(</mo><mi>d</mi><mo>)</mo><mo>,</mo><mi>W</mi><mo>(</mo><mi>d</mi><mo>)</mo><mo>)</mo></mrow></semantics></math></inline-formula>, and call the two components <i>area</i> and <i>width</i> (respectively). The main result of this paper is the following: the rectangle-area formula <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>(</mo><mi>d</mi><mo>)</mo><mo>=</mo><mi>L</mi><mo>(</mo><mi>d</mi><mo>)</mo><mi>W</mi><mo>(</mo><mi>d</mi><mo>)</mo></mrow></semantics></math></inline-formula> holds for any Dirichlet polynomial <i>d</i>. In other words, the entropy of an empirical distribution can be calculated entirely in terms of the homomorphism <i>h</i> applied to its corresponding Dirichlet polynomial. We also show that similar results hold for the cross entropy.
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