Harmonic entropy: Difference between revisions

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Lastly, it so happens that it will be much easier to understand our analytic continuation if we look at the exponential of the UHE, times <math>(1-a)</math>, rather than the UHE itself. The reasons for this will become clear later. If we do so, we get

Lastly, it so happens that it will be much easier to understand our analytic continuation if we look at the exponential of the UHE times <math>(1-a)</math>, rather than the UHE itself. The reasons for this will become clear later. If we do so, we get

<math>\displaystyle \exp((1-a) \text{UHE}_a(c)) = \left( S^a \ast K^a \right)(-c)</math>

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In our original derivation of the analytic continuation, we temporarily changed the weighting for rationals from <math>(nd)^{0.5}</math> to some other <math>(nd)^w</math>, with <math>w > 1</math>, for the sake of obtaining a series that converges. We then changed the exponent back to <math>0.5</math>.

This can be thought of as giving us another free parameter to HE, in addition to <math>s</math> and <math>a</math>: the exponent for the weighting for each rational. That is, although Paul originally derived the <math>(nd)^0.5</math> exponent empirically by studying the behavior of mediant-to-mediant HE for Tenney-bounded rationals, there is no reason we can't simply that exponent to something else. As shown before, so long as that exponent is greater than 1, unnormalized HE will converge in the limit as <math>N -> \infty</math>, and will converge to the same thing whether we are bounding <math>nd < N</math>, <math>\max(n,d) < N</math>, or anything else (see again [https://math.stackexchange.com/questions/2593993/convergence-of-product-of-series-to-zeta-function here]). We can then analytically continue to the case where <math>w < 1</math>.

This can be thought of as giving us another free parameter to HE, in addition to <math>s</math> and <math>a</math>: the exponent for the weighting for each rational. That is, although Paul originally derived the <math>(nd)^{0.5}</math> exponent empirically by studying the behavior of mediant-to-mediant HE for Tenney-bounded rationals, there is no reason we can't simply that exponent to something else. As shown before, so long as that exponent is greater than 1, unnormalized HE will converge in the limit as <math>N -> \infty</math>, and will converge to the same thing whether we are bounding <math>nd < N</math>, <math>\max(n,d) < N</math>, or anything else (see again [https://math.stackexchange.com/questions/2593993/convergence-of-product-of-series-to-zeta-function here]). We can then analytically continue to the case where <math>w < 1</math>.

If we add this as a third parameter, called <math>w</math> we can modify our definition of exp-UHE as follows:

@@ Line 422: / Line 422: @@
-Lastly, it so happens that it will be much easier to understand our analytic continuation if we look at the exponential of the UHE, times <math>(1-a)</math>, rather than the UHE itself. The reasons for this will become clear later. If we do so, we get
+Lastly, it so happens that it will be much easier to understand our analytic continuation if we look at the exponential of the UHE times <math>(1-a)</math>, rather than the UHE itself. The reasons for this will become clear later. If we do so, we get
 <math>\displaystyle \exp((1-a) \text{UHE}_a(c)) = \left( S^a \ast K^a \right)(-c)</math>
@@ Line 624: / Line 624: @@
 In our original derivation of the analytic continuation, we temporarily changed the weighting for rationals from <math>(nd)^{0.5}</math> to some other <math>(nd)^w</math>, with <math>w > 1</math>, for the sake of obtaining a series that converges. We then changed the exponent back to <math>0.5</math>.
-This can be thought of as giving us another free parameter to HE, in addition to <math>s</math> and <math>a</math>: the exponent for the weighting for each rational. That is, although Paul originally derived the <math>(nd)^0.5</math> exponent empirically by studying the behavior of mediant-to-mediant HE for Tenney-bounded rationals, there is no reason we can't simply that exponent to something else. As shown before, so long as that exponent is greater than 1, unnormalized HE will converge in the limit as <math>N -> \infty</math>, and will converge to the same thing whether we are bounding <math>nd < N</math>, <math>\max(n,d) < N</math>, or anything else (see again [https://math.stackexchange.com/questions/2593993/convergence-of-product-of-series-to-zeta-function here]). We can then analytically continue to the case where <math>w < 1</math>.
+This can be thought of as giving us another free parameter to HE, in addition to <math>s</math> and <math>a</math>: the exponent for the weighting for each rational. That is, although Paul originally derived the <math>(nd)^{0.5}</math> exponent empirically by studying the behavior of mediant-to-mediant HE for Tenney-bounded rationals, there is no reason we can't simply that exponent to something else. As shown before, so long as that exponent is greater than 1, unnormalized HE will converge in the limit as <math>N -> \infty</math>, and will converge to the same thing whether we are bounding <math>nd < N</math>, <math>\max(n,d) < N</math>, or anything else (see again [https://math.stackexchange.com/questions/2593993/convergence-of-product-of-series-to-zeta-function here]). We can then analytically continue to the case where <math>w < 1</math>.
 If we add this as a third parameter, called <math>w</math> we can modify our definition of exp-UHE as follows: