The Riemann zeta function and tuning: Difference between revisions
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<math>\sum_2^\infty \frac{||x \log_2 q||^2}{q^s}</math> | <math>\sum_2^\infty \frac{||x \log_2 q||^2}{q^s}</math> | ||
If s is greater than one, this does converge. However, we might want to make a few adjustments. For one thing, if the error is low enough that the tuning is consistent, then the error of the square of a prime is twice that of the prime, of the cube tripled, and so forth until the error becomes inconsistent. When the weighting uses logarithms and error measures are consistent, then the logarithmic weighting cancels this effect out, so we might consider that prime powers were implicitly included in the Tenney-Euclidean measure. We can go ahead and include them by adding a factor of 1/n for each prime power p^n. A somewhat peculiar but useful way to write the result of doing this is in terms of the [ | If s is greater than one, this does converge. However, we might want to make a few adjustments. For one thing, if the error is low enough that the tuning is consistent, then the error of the square of a prime is twice that of the prime, of the cube tripled, and so forth until the error becomes inconsistent. When the weighting uses logarithms and error measures are consistent, then the logarithmic weighting cancels this effect out, so we might consider that prime powers were implicitly included in the Tenney-Euclidean measure. We can go ahead and include them by adding a factor of 1/n for each prime power p^n. A somewhat peculiar but useful way to write the result of doing this is in terms of the [[Wikipedia:Von Mangoldt function|Von Mangoldt function]], an [[Wikipedia:arithmetic function|arithmetic function]] on positive integers which is equal to ln p on prime powers p^n, and is zero elsewhere. This is written using a capital lambda, as Λ(n), and in terms of it we can include prime powers in our error function as | ||
<math>\sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{||x \log_2 n||^2}{n^s}</math> | <math>\sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{||x \log_2 n||^2}{n^s}</math> | ||
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where the summation is taken formally over all positive integers, though only the primes and prime powers make a nonzero contribution. | where the summation is taken formally over all positive integers, though only the primes and prime powers make a nonzero contribution. | ||
Another consequence of the above definition which might be objected to is that it results in a function with a [ | Another consequence of the above definition which might be objected to is that it results in a function with a [[Wikipedia:Continuous_function#Relation_to_differentiability_and_integrability|discontinuous derivative]], whereas a smooth function be preferred. The function ||x||^2 is quadratically increasing near integer values of x, and is periodic with period 1. Another function with these same properties is 1 - cos(2πx), which is a smooth and in fact an [[Wikipedia:entire function|entire function]]. Let us therefore now define for any s > 1 | ||
<math>E_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{1 - \cos(2 \pi x \log_2 n)}{n^s}</math> | <math>E_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{1 - \cos(2 \pi x \log_2 n)}{n^s}</math> | ||
For any fixed s > 1 this gives a [ | For any fixed s > 1 this gives a real [[Wikipedia:analytic function|analytic function]] defined for all x, and hence with all the smoothness properties we could desire. | ||
We can clean up this definition to get essentially the same function: | We can clean up this definition to get essentially the same function: | ||
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<math>F_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{\cos(2 \pi x \log_2 n)}{n^s}</math> | <math>F_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{\cos(2 \pi x \log_2 n)}{n^s}</math> | ||
This new function has the property that <math>F_s(x) = F_s(0) - E_s(x)</math>, so that all we have done is flip the sign of <math>E_s(x)</math> and offset it vertically. This now increases to a maximum value for low errors, rather than declining to a minimum. Of more interest is the fact that it is a known mathematical function, which can be expressed in terms of the real part of the logarithm of the [ | This new function has the property that <math>F_s(x) = F_s(0) - E_s(x)</math>, so that all we have done is flip the sign of <math>E_s(x)</math> and offset it vertically. This now increases to a maximum value for low errors, rather than declining to a minimum. Of more interest is the fact that it is a known mathematical function, which can be expressed in terms of the real part of the logarithm of the [[Wikipedia:Riemann zeta function|Riemann zeta function]]: | ||
<math>F_s(x) = \Re \ln \zeta(s + 2 \pi i x/\ln 2)</math> | <math>F_s(x) = \Re \ln \zeta(s + 2 \pi i x/\ln 2)</math> | ||
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So long as s is greater than or equal to one, the absolute value of the zeta function can be seen as a relative error measurement. However, the rationale for that view of things departs when s is less than one, particularly in the [http://mathworld.wolfram.com/CriticalStrip.html critical strip], when s lies between zero and one. As s approaches the value s=1/2 of the [http://mathworld.wolfram.com/CriticalLine.html critical line], the information content, so to speak, of the zeta function concerning higher primes increases and it behaves increasingly like a badness measure (or more correctly, since we have inverted it, like a goodness measure.) The quasi-symmetric [http://planetmath.org/encyclopedia/FunctionalEquationOfTheRiemannZetaFunction.html functional equation] of the zeta function tells us that past the critical line the information content starts to decrease again, with 1-s and s having the same information content. Hence it is the zeta function between s=1/2 and s=1, and especially the zeta function along the critical line s=1/2, which is of the most interest. | So long as s is greater than or equal to one, the absolute value of the zeta function can be seen as a relative error measurement. However, the rationale for that view of things departs when s is less than one, particularly in the [http://mathworld.wolfram.com/CriticalStrip.html critical strip], when s lies between zero and one. As s approaches the value s=1/2 of the [http://mathworld.wolfram.com/CriticalLine.html critical line], the information content, so to speak, of the zeta function concerning higher primes increases and it behaves increasingly like a badness measure (or more correctly, since we have inverted it, like a goodness measure.) The quasi-symmetric [http://planetmath.org/encyclopedia/FunctionalEquationOfTheRiemannZetaFunction.html functional equation] of the zeta function tells us that past the critical line the information content starts to decrease again, with 1-s and s having the same information content. Hence it is the zeta function between s=1/2 and s=1, and especially the zeta function along the critical line s=1/2, which is of the most interest. | ||
As s>1 gets larger, the Dirichlet series for the zeta function is increasingly dominated by the 2 term, getting ever closer to simply 1 + 2^(-z), which approaches 1 as s = Re(z) becomes larger. When s >> 1 and x is an integer, the real part of zeta is approximately 1 + 2^(-s), and the imaginary part is approximately zero; that is, zeta is approximately real. Starting from s = +∞ with x an integer, we can trace a line back towards the critical strip on which zeta is real. Since when s >> 1 the derivative is approximately -ln(2)/2^s, it is negative on this line of real values for zeta, meaning that the real value for zeta increases as s decreases. The zeta function approaches 1 uniformly as s increases to infinity, so as s decreases, the real-valued zeta function along this line of real values continues to increase though all real values from 1 to infinity monotonically. When it crosses the critical line where s=1/2, it produces a real value of zeta on the critical line. Points on the critical line where ζ(1/2 + i g) are real are called "Gram points", after [ | As s>1 gets larger, the Dirichlet series for the zeta function is increasingly dominated by the 2 term, getting ever closer to simply 1 + 2^(-z), which approaches 1 as s = Re(z) becomes larger. When s >> 1 and x is an integer, the real part of zeta is approximately 1 + 2^(-s), and the imaginary part is approximately zero; that is, zeta is approximately real. Starting from s = +∞ with x an integer, we can trace a line back towards the critical strip on which zeta is real. Since when s >> 1 the derivative is approximately -ln(2)/2^s, it is negative on this line of real values for zeta, meaning that the real value for zeta increases as s decreases. The zeta function approaches 1 uniformly as s increases to infinity, so as s decreases, the real-valued zeta function along this line of real values continues to increase though all real values from 1 to infinity monotonically. When it crosses the critical line where s=1/2, it produces a real value of zeta on the critical line. Points on the critical line where ζ(1/2 + i g) are real are called "Gram points", after [[Wikipedia:Jørgen Pedersen Gram|Jørgen Pedersen Gram]]. We thus have associated pure-octave edos, where x is an integer, to a value near to the pure octave, at the special sorts of Gram points which corresponds to edos. | ||
Because the value of zeta increased continuously as it made its way from +∞ to the critical line, we might expect the values of zeta at these special Gram points to be relatively large. This would be especially true if -ζ'(z) is getting a boost from other small primes as it travels toward the Gram point. A complex formula due to [ | Because the value of zeta increased continuously as it made its way from +∞ to the critical line, we might expect the values of zeta at these special Gram points to be relatively large. This would be especially true if -ζ'(z) is getting a boost from other small primes as it travels toward the Gram point. A complex formula due to [[Wikipedia:Bernhard Riemann|Bernhard Riemann]] which he failed to publish because it was so nasty becomes a bit simpler when used at a Gram point. It is named the [[Wikipedia:Riemann-Siegel formula|Riemann-Siegel formula]] since [[Wikipedia:Carl Ludwig Siegel|Carl Ludwig Siegel]] went looking for it and was able to reconstruct it after rooting industriously around in Riemann's unpublished papers. From this formula, it is apparent that when x corresponds to a good edo, the value of ζ(1/2 + i g) at the corresponding Gram point should be especially large. | ||
=The Z function= | =The Z function= | ||
The absolute value ζ(1/2 + i g) at a Gram point corresponding to an edo is near to a local maximum, but not actually at one. At the local maximum, of course, the partial derivative of ζ(1/2 + i t) with respect to t will be zero; however this does not mean its derivative there will be zero. In fact, the [ | The absolute value ζ(1/2 + i g) at a Gram point corresponding to an edo is near to a local maximum, but not actually at one. At the local maximum, of course, the partial derivative of ζ(1/2 + i t) with respect to t will be zero; however this does not mean its derivative there will be zero. In fact, the [[Wikipedia:Riemann hypothesis|Riemann hypothesis]] is equivalent to the claim that all zeros of ζ'(s + i t) occur when s > 1/2, which is where all known zeros lie. These do not have values of t corresponding to good edos. For this and other reasons, it is helpful to have a function which is real for values on the critical line but whose absolute value is the same as that of zeta. This is provided by the [[Wikipedia:Z function|Z function]]. | ||
In order to define the Z function, we need first to define the [ | In order to define the Z function, we need first to define the [[Wikipedia:Riemann-Siegel theta function|Riemann-Siegel theta function]], and in order to do that, we first need to define the [http://mathworld.wolfram.com/LogGammaFunction.html Log Gamma function]. This is not defined as the natural log of the Gamma function since that has a more complicated branch cut structure; instead, the principal branch of the Log Gamma function is defined as having a branch cut along the negative real axis, and is given by the series | ||
<math>\Upsilon(z) = -\gamma z - \ln z + \sum_{k=1}^\infty \frac{z}{k} - \ln(1 + \frac{z}{k})</math> | <math>\Upsilon(z) = -\gamma z - \ln z + \sum_{k=1}^\infty \frac{z}{k} - \ln(1 + \frac{z}{k})</math> | ||
where the lower-case gamma is [ | where the lower-case gamma is [[Wikipedia:Euler's constant|Euler's constant]]. We now may define the Riemann-Siegel theta function as | ||
<math>\theta(z) = (\Upsilon((1 + 2 i z)/4) - \Upsilon((1 - 2 i z)/4))/(2 i) - \ln(\pi) z/2</math> | <math>\theta(z) = (\Upsilon((1 + 2 i z)/4) - \Upsilon((1 - 2 i z)/4))/(2 i) - \ln(\pi) z/2</math> | ||
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Since the arctangent function is holomorphic in the strip with imaginary part between -1 and 1, it follows from the above formula, or arguing from the previous one, that theta is holomorphic in the strip with imaginary part between -1/2 and 1/2. It may be described for real arguments as an odd real analytic function of x, increasing when |x| > 6.29. Plots of it may be studied by use of the Wolfram [http://functions.wolfram.com/webMathematica/FunctionPlotting.jsp?name=RiemannSiegelTheta online function plotter]. | Since the arctangent function is holomorphic in the strip with imaginary part between -1 and 1, it follows from the above formula, or arguing from the previous one, that theta is holomorphic in the strip with imaginary part between -1/2 and 1/2. It may be described for real arguments as an odd real analytic function of x, increasing when |x| > 6.29. Plots of it may be studied by use of the Wolfram [http://functions.wolfram.com/webMathematica/FunctionPlotting.jsp?name=RiemannSiegelTheta online function plotter]. | ||
Using the theta and zeta functions, we define the [ | Using the theta and zeta functions, we define the [[Wikipedia:Z function|Z function]] as | ||
<math>Z(t) = \exp(i \theta(t)) \zeta(1/2 + it)</math> | <math>Z(t) = \exp(i \theta(t)) \zeta(1/2 + it)</math> | ||
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Using the [http://functions.wolfram.com/webMathematica/FunctionPlotting.jsp?name=RiemannSiegelZ online plotter] we can plot Z in the regions corresponding to scale divisions, using the conversion factor t = 2π/ln(2) x, for x a number near or at an edo number. Hence, for instance, to plot 12 plot around 108.777, to plot 31 plot around 281.006, and so forth. An alternative plotter is the applet [http://web.viu.ca/pughg/RiemannZeta/RiemannZetaLong.html here]. | Using the [http://functions.wolfram.com/webMathematica/FunctionPlotting.jsp?name=RiemannSiegelZ online plotter] we can plot Z in the regions corresponding to scale divisions, using the conversion factor t = 2π/ln(2) x, for x a number near or at an edo number. Hence, for instance, to plot 12 plot around 108.777, to plot 31 plot around 281.006, and so forth. An alternative plotter is the applet [http://web.viu.ca/pughg/RiemannZeta/RiemannZetaLong.html here]. | ||
If you have access to [ | If you have access to [[Wikipedia:Mathematica|Mathematica]], which has Z, zeta and theta as a part of its suite of initially defined functions, you can do even better. Below is a Mathematicia-generated plot of Z(2πx/ln(2)) in the region around 12edo: | ||
[[File:plot12.png|alt=plot12.png|plot12.png]] | [[File:plot12.png|alt=plot12.png|plot12.png]] | ||
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=Zeta EDO lists= | =Zeta EDO lists= | ||
If we examine the increasingly larger peak values of |Z(x)|, we find they occur with values of x such that Z'(x) = 0 near to integers, so that there is a sequence of [[EDO|edo]]s [[1edo|1]], [[2edo|2]], [[3edo|3]], [[4edo|4]], [[5edo|5]], [[7edo|7]], [[10edo|10]], [[12edo|12]], [[19edo|19]], [[22edo|22]], [[27edo|27]], [[31edo|31]], [[41edo|41]], [[53edo|53]], [[72edo|72]], [[99edo|99]], [[118edo|118]], [[130edo|130]], [[152edo|152]], [[171edo|171]], [[217edo|217]], [[224edo|224]], [[270edo|270]], [[342edo|342]], [[422edo|422]], [[441edo|441]], [[494edo|494]], [[742edo|742]], [[764edo|764]], [[935edo|935]], [[954edo|954]], [[1012edo|1012]], [[1106edo|1106]], [[1178edo|1178]], [[1236edo|1236]], [[1395edo|1395]], [[1448edo|1448]], [[1578edo|1578]], [[2460edo|2460]], [[2684edo|2684]], [[3395edo|3395]], [[5585edo|5585]], [[6079edo|6079]], [[7033edo|7033]], [[8269edo|8269]], [[8539edo|8539]], [[11664edo|11664]], [[14348edo|14348]], [[16808edo|16808]], [[28742edo|28742]], [[34691edo|34691]] ... of ''zeta peak edos''. This is listed in the On-Line Encyclopedia of Integer Sequences as | If we examine the increasingly larger peak values of |Z(x)|, we find they occur with values of x such that Z'(x) = 0 near to integers, so that there is a sequence of [[EDO|edo]]s [[1edo|1]], [[2edo|2]], [[3edo|3]], [[4edo|4]], [[5edo|5]], [[7edo|7]], [[10edo|10]], [[12edo|12]], [[19edo|19]], [[22edo|22]], [[27edo|27]], [[31edo|31]], [[41edo|41]], [[53edo|53]], [[72edo|72]], [[99edo|99]], [[118edo|118]], [[130edo|130]], [[152edo|152]], [[171edo|171]], [[217edo|217]], [[224edo|224]], [[270edo|270]], [[342edo|342]], [[422edo|422]], [[441edo|441]], [[494edo|494]], [[742edo|742]], [[764edo|764]], [[935edo|935]], [[954edo|954]], [[1012edo|1012]], [[1106edo|1106]], [[1178edo|1178]], [[1236edo|1236]], [[1395edo|1395]], [[1448edo|1448]], [[1578edo|1578]], [[2460edo|2460]], [[2684edo|2684]], [[3395edo|3395]], [[5585edo|5585]], [[6079edo|6079]], [[7033edo|7033]], [[8269edo|8269]], [[8539edo|8539]], [[11664edo|11664]], [[14348edo|14348]], [[16808edo|16808]], [[28742edo|28742]], [[34691edo|34691]] ... of ''zeta peak edos''. This is listed in the On-Line Encyclopedia of Integer Sequences as {{OEIS|A117536}}. | ||
Similarly, if we take the integral of |Z(x)| between successive zeros, and use this to define a sequence of increasing values for this integral, these again occur near integers and define an edo. This sequence, the ''zeta integral edos'', goes [[2edo|2]], [[5edo|5]], [[7edo|7]], [[12edo|12]], [[19edo|19]], [[31edo|31]], [[41edo|41]], [[53edo|53]], [[72edo|72]], [[130edo|130]], [[171edo|171]], [[224edo|224]], [[270edo|270]], [[764edo|764]], [[954edo|954]], [[1178edo|1178]], [[1395edo|1395]], [[1578edo|1578]], [[2684edo|2684]], [[3395edo|3395]], [[7033edo|7033]], [[8269edo|8269]], [[8539edo|8539]], [[14348edo|14348]], [[16808edo|16808]], [[36269edo|36269]], [[58973edo|58973]] ... This is listed in the OEIS as | Similarly, if we take the integral of |Z(x)| between successive zeros, and use this to define a sequence of increasing values for this integral, these again occur near integers and define an edo. This sequence, the ''zeta integral edos'', goes [[2edo|2]], [[5edo|5]], [[7edo|7]], [[12edo|12]], [[19edo|19]], [[31edo|31]], [[41edo|41]], [[53edo|53]], [[72edo|72]], [[130edo|130]], [[171edo|171]], [[224edo|224]], [[270edo|270]], [[764edo|764]], [[954edo|954]], [[1178edo|1178]], [[1395edo|1395]], [[1578edo|1578]], [[2684edo|2684]], [[3395edo|3395]], [[7033edo|7033]], [[8269edo|8269]], [[8539edo|8539]], [[14348edo|14348]], [[16808edo|16808]], [[36269edo|36269]], [[58973edo|58973]] ... This is listed in the OEIS as {{OEIS|A117538}}. The zeta integral edos seem to be, on the whole, the best of the zeta function sequences, but the other two should not be discounted; the peak values seem to give more weight to the lower primes, and the zeta gap sequence discussed below to the higher primes. | ||
Finally, taking the midpoints of the successively larger normalized gaps between the zeros of Z leads to a list of ''zeta gap edos''. These are [[2edo|2]], [[3edo|3]], [[5edo|5]], [[7edo|7]], [[12edo|12]], [[19edo|19]], [[31edo|31]], [[46edo|46]], [[53edo|53]], [[72edo|72]], [[270edo|270]], [[311edo|311]], [[954edo|954]], [[1178edo|1178]], [[1308edo|1308]], [[1395edo|1395]], [[1578edo|1578]], [[3395edo|3395]], [[4190edo|4190]] ... Since the density of the zeros increases logarithmically, the normalization is to divide through by the log of the midpoint. These edos are listed in the OEIS as | Finally, taking the midpoints of the successively larger normalized gaps between the zeros of Z leads to a list of ''zeta gap edos''. These are [[2edo|2]], [[3edo|3]], [[5edo|5]], [[7edo|7]], [[12edo|12]], [[19edo|19]], [[31edo|31]], [[46edo|46]], [[53edo|53]], [[72edo|72]], [[270edo|270]], [[311edo|311]], [[954edo|954]], [[1178edo|1178]], [[1308edo|1308]], [[1395edo|1395]], [[1578edo|1578]], [[3395edo|3395]], [[4190edo|4190]] ... Since the density of the zeros increases logarithmically, the normalization is to divide through by the log of the midpoint. These edos are listed in the OEIS as {{OEIS|A117537}}. The zeta gap edos seem to weight higher primes more heavily and have the advantage of being easy to compute from a table of zeros on the critical line. | ||
=Optimal Octave Stretch= | =Optimal Octave Stretch= | ||
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=The Black Magic Formulas= | =The Black Magic Formulas= | ||
When [[Gene_Ward_Smith|Gene Smith]] discovered these formulas in the 70s, he thought of them as "black magic" formulas not because of any aura of evil, but because they seemed mysteriously to give you something for next to nothing. They are based on Gram points and the Riemann-Siegel theta function θ(t). Recall that a Gram point is a point on the critical line where ζ(1/2 + ig) is real. This implies that exp(iθ(g)) is real, so that θ(g)/π is an integer. Theta has an [ | When [[Gene_Ward_Smith|Gene Smith]] discovered these formulas in the 70s, he thought of them as "black magic" formulas not because of any aura of evil, but because they seemed mysteriously to give you something for next to nothing. They are based on Gram points and the Riemann-Siegel theta function θ(t). Recall that a Gram point is a point on the critical line where ζ(1/2 + ig) is real. This implies that exp(iθ(g)) is real, so that θ(g)/π is an integer. Theta has an [[Wikipedia:asymptotic expansion|asymptotic expansion]] | ||
<math>\theta(t) \sim \frac{t}{2}\log \frac{t}{2\pi} - \frac{t}{2} - \frac{\pi}{8}+\frac{1}{48t}+ \frac{7}{5760t^3}+\cdots</math> | <math>\theta(t) \sim \frac{t}{2}\log \frac{t}{2\pi} - \frac{t}{2} - \frac{\pi}{8}+\frac{1}{48t}+ \frac{7}{5760t^3}+\cdots</math> | ||
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=Computing zeta= | =Computing zeta= | ||
There are various approaches to the question of computing the zeta function, but perhaps the simplest is the use of the [ | There are various approaches to the question of computing the zeta function, but perhaps the simplest is the use of the [[Wikipedia:Dirichlet eta function|Dirichlet eta function]] which was introduced to mathematics by [[Wikipedia:Johann Peter Gustav Lejeune Dirichlet|Johann Peter Gustav Lejeune Dirichlet]], who despite his name was a German and the brother-in-law of [[Wikipedia:Felix Mendelssohn|Felix Mendelssohn]]. | ||
The zeta function has a [http://mathworld.wolfram.com/SimplePole.html simple pole] at z=1 which forms a barrier against continuing it with its [ | The zeta function has a [http://mathworld.wolfram.com/SimplePole.html simple pole] at z=1 which forms a barrier against continuing it with its [[Wikipedia:Euler product|Euler product]] or [[Wikipedia:Dirichlet series|Dirichlet series]] representation. We could subtract off the pole, or multiply by a factor of (z-1), but at the expense of losing the character of a Dirichlet series or Euler product. A better method is to multiply by a factor of (1-2^(1-z)), leading to the eta function: | ||
<math>\eta(z) = (1-2^{1-z})\zeta(z) = \sum_{n=1}^\infty (-1)^{n-1} n^{-z} | <math>\eta(z) = (1-2^{1-z})\zeta(z) = \sum_{n=1}^\infty (-1)^{n-1} n^{-z} | ||
= \frac{1}{1^z} - \frac{1}{2^z} + \frac{1}{3^z} - \frac{1}{4^z} + \cdots</math> | = \frac{1}{1^z} - \frac{1}{2^z} + \frac{1}{3^z} - \frac{1}{4^z} + \cdots</math> | ||
The Dirichlet series for the zeta function is absolutely convergent when s>1, justifying the rearrangement of terms leading to the alternating series for eta, which converges conditionally in the critical strip. The extra factor introduces zeros of the eta function at the points 1 + 2πix/ln(2) corresponding to pure octave divisions along the line s=1, but no other zeros, and in particular none in the critical strip and along the critical line. The convergence of the alternating series can be greatly accelerated by applying [ | The Dirichlet series for the zeta function is absolutely convergent when s>1, justifying the rearrangement of terms leading to the alternating series for eta, which converges conditionally in the critical strip. The extra factor introduces zeros of the eta function at the points 1 + 2πix/ln(2) corresponding to pure octave divisions along the line s=1, but no other zeros, and in particular none in the critical strip and along the critical line. The convergence of the alternating series can be greatly accelerated by applying [[Wikipedia:Euler summation|Euler summation]]. | ||
=Relationship to Harmonic Entropy= | =Relationship to Harmonic Entropy= | ||