Harmonic entropy: Difference between revisions

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The only technical caveat is that we use the HE of the "unnormalized" probability distribution. However, in the large limit of ''N'', this appears to agree closely with the usual HE. We go into more detail below about this.

Our basic approach is: rather than weighting intervals by (''nd'')0.5, we choose a different exponent, such as (''nd'')2. For an exponent which is large enough (we will show that it must be greater than 1), HE does indeed converge as {{nowrap|''N'' → ∞}}, 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 (''nd'')0.5 weighting, which we then show empirically does indeed appear to be what HE converges on for large values of ''N''.

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

In short, what we will show is that the Fourier Transform of this unnormalized harmonic Shannon entropy is given by

Line 430:

It would be nice to show the exact relationship of unnormalized entropy to the normalized entropy in the limit of large ''N'', and whether the two converge to be exactly equal (perhaps given some miniscule adjustment in ''s'' or ''a''). However, we will leave this for future research, as well as the question of how to do an exact derivation of normalized HE.

For now, we will start with a derivation of the unnormalized entropy for {{nowrap|''N'' {{=}} ∞}}, as an interesting function worthy of study in its own right—not only because it looks exactly like HE, but because it leads to an expression for unnormalized HE in terms of the [[~~The Riemann zeta function and tuning|~~Riemann zeta function]].

For now, we will start with a derivation of the unnormalized entropy for {{nowrap|''N'' {{=}} ∞}}, as an interesting function worthy of study in its own right—not only because it looks exactly like HE, but because it leads to an expression for unnormalized HE in terms of the [[Riemann zeta function]].

=== Derivation ===

@@ Line 401: / Line 401: @@
 The only technical caveat is that we use the HE of the "unnormalized" probability distribution. However, in the large limit of ''N'', this appears to agree closely with the usual HE. We go into more detail below about this.
-Our basic approach is: rather than weighting intervals by (''nd'')<sup>0.5</sup>, we choose a different exponent, such as (''nd'')<sup>2</sup>. For an exponent which is large enough (we will show that it must be greater than 1), HE does indeed converge as {{nowrap|''N'' → ∞}}, 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 (''nd'')<sup>0.5</sup> weighting, which we then show empirically does indeed appear to be what HE converges on for large values of ''N''.
+Our basic approach is: rather than weighting intervals by (''nd'')<sup>0.5</sup>, we choose a different exponent, such as (''nd'')<sup>2</sup>. For an exponent which is large enough (we will show that it must be greater than 1), HE does indeed converge as {{nowrap|''N'' → ∞}}, and we show that this yields an expression related to the [[Riemann zeta function]]. We can then use the analytic continuation of the zeta function to obtain an analytically continued curve for the (''nd'')<sup>0.5</sup> weighting, which we then show empirically does indeed appear to be what HE converges on for large values of ''N''.
 In short, what we will show is that the Fourier Transform of this unnormalized harmonic Shannon entropy is given by
@@ Line 430: / Line 430: @@
 It would be nice to show the exact relationship of unnormalized entropy to the normalized entropy in the limit of large ''N'', and whether the two converge to be exactly equal (perhaps given some miniscule adjustment in ''s'' or ''a''). However, we will leave this for future research, as well as the question of how to do an exact derivation of normalized HE.
-For now, we will start with a derivation of the unnormalized entropy for {{nowrap|''N'' {{=}} ∞}}, as an interesting function worthy of study in its own right—not only because it looks exactly like HE, but because it leads to an expression for unnormalized HE in terms of the [[The Riemann zeta function and tuning|Riemann zeta function]].
+For now, we will start with a derivation of the unnormalized entropy for {{nowrap|''N'' {{=}} ∞}}, as an interesting function worthy of study in its own right—not only because it looks exactly like HE, but because it leads to an expression for unnormalized HE in terms of the [[Riemann zeta function]].
 === Derivation ===