Harmonic entropy: Difference between revisions

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Our basic approach is: rather than weighting intervals by <math>(nd)^{0.5}</math>, we choose a different exponent, such as <math>(nd)^2</math>. For an exponent which is large enough (we will show that it must be greater than 1), HE does indeed converge as <math>N \to \infty</math>, and we show that this yields an expression related to the [[The_Riemann_Zeta_Function_and_Tuning|Riemann Zeta function]]. We can then use the analytic continuation of the zeta function to obtain an analytically continued curve for the <math>(nd)^{0.5}</math> weighting, which we then show empirically does indeed appear to be what HE converges on for large values of <math>N</math>.

In short, what we will show is that the Fourier Transform of Harmonic Shannon Entropy is given by

<math>|\zeta(0.5+it)|^2 \cdot \overline {\phi(t)}</math>

where <math>\phi(t)</math> is the characteristic function of the spreading distribution and <math>\overline {\phi(t)}</math> is complex conjugation. Below we also give an expression for the Renyi entropy for arbitrary choice of the parameter <math>a</math>.

This enables us to speak cognizantly of the harmonic entropy of an interval as measured against ''all'' rational numbers.

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 Our basic approach is: rather than weighting intervals by <math>(nd)^{0.5}</math>, we choose a different exponent, such as <math>(nd)^2</math>. For an exponent which is large enough (we will show that it must be greater than 1), HE does indeed converge as <math>N \to \infty</math>, and we show that this yields an expression related to the [[The_Riemann_Zeta_Function_and_Tuning|Riemann Zeta function]]. We can then use the analytic continuation of the zeta function to obtain an analytically continued curve for the <math>(nd)^{0.5}</math> weighting, which we then show empirically does indeed appear to be what HE converges on for large values of <math>N</math>.
+In short, what we will show is that the Fourier Transform of Harmonic Shannon Entropy is given by
+<math>|\zeta(0.5+it)|^2 \cdot \overline {\phi(t)}</math>
+where <math>\phi(t)</math> is the characteristic function of the spreading distribution and <math>\overline {\phi(t)}</math> is complex conjugation. Below we also give an expression for the Renyi entropy for arbitrary choice of the parameter <math>a</math>.
 This enables us to speak cognizantly of the harmonic entropy of an interval as measured against ''all'' rational numbers.