Riemann zeta function: Difference between revisions

Line 41:

You may also view the graph of zeta along the critical line on Desmos: [https://www.desmos.com/calculator/dstp7wnidf Zeta in Desmos]. This makes it easier to see peaks, but only works for sigma=1/2.

=== Plots ===

Below are some demonstrative plots of the zeta function (strictly speaking, the [[#The Z function: a mathematically convenient version of zeta|Z function]]) on the critical line.

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 {{nowrap|''t'' {{=}} {{sfrac|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 {{w|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 Mathematica-generated plot of Z({{frac|2π''x''|ln(2)}}) in the region around 12edo:

[[File:plot12.png|alt=plot12.png|plot12.png]]

The peak around 12 is both higher and wider than the local maximums above 11 and 13, indicating its superiority as an edo. Note also that the peak occurs at a point slightly larger than 12; this indicates the octave is slightly compressed in the zeta tuning for 12. The size of a step in octaves is 1/''x'', and hence the size of the octave in the zeta peak value tuning for ''N''edo is ''N''/''x''; if ''x'' is slightly larger than ''N'' as here with {{nowrap|''N'' {{=}} 12}}, the size of the zeta tuned octave will be slightly less than a pure octave. Similarly, when the peak occurs with ''x'' less than ''N'', we have stretched octaves.

For larger edos, the width of the peak narrows, but for strong edos the height more than compensates, measured in terms of the area under the peak (the absolute value of the integral of Z between two zeros.) Note how 270 completely dominates its neighbors:

[[File:plot270.png|alt=plot270.png|plot270.png]]

Note that for one of its neighbors, 271, it isn't entirely clear which peak value corresponds to the line of real values from +∞. This can be determined by looking at the absolute value of zeta along other ''s'' values, such as {{nowrap|''s'' {{=}} 1}} or {{nowrap|''s'' {{=}} {{sfrac|3|4}}}}, and in this case the local minimum at 271.069 is the value in question. However, other peak values are not without their interest; the local maximum at 270.941, for instance, is associated to a different mapping for 3.

To generate this plot using the free version of Wolfram Cloud, you can run <code>Plot[Abs[RiemannSiegelZ[9.06472028x]], {x, 11.9, 12.1}]</code> and then in the menu select '''Evaluation > Evaluate Cells'''. Change "'''11.9'''" and "'''12.1'''" to whatever values you want, e.g. to view the curve around 15edo you might use the values "'''14.9'''" and "'''15.1'''".

You can also view the plot using [https://www.desmos.com/calculator/dstp7wnidf Zeta in Desmos].

== Gene Smith's original derivation ==

~~=== The relation of zeta to tuning ===~~

Suppose ''x'' is a variable representing some equal division of the octave. For example, if {{nowrap|''x'' {{=}} 80}}, ''x'' reflects 80edo with a step size of 15 cents and with pure octaves. Suppose that ''x'' can also be continuous, so that it can also represent fractional or "nonoctave" divisions as well. The [[Bohlen–Pierce scale]], 13 equal divisions of 3/1, is approximately 8.202 equal divisions of the "octave" (although the octave itself does not appear in this tuning), and would hence be represented by a value of {{nowrap|''x'' {{=}} 8.202}}.

Line 116:

Line 136:

so that we see that the absolute value of the zeta function serves to measure the relative error of an equal division.

~~=== The critical strip, zeta peaks, and Gram points ===~~

So long as {{nowrap|''s'' ≥ 1}}, 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 {{nowrap|''s'' < 1}}, particularly in the [http://mathworld.wolfram.com/CriticalStrip.html critical strip], when {{nowrap|0 < ''s'' < 1}}. As s approaches the value {{nowrap|''s'' {{=}} {{sfrac|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 [https://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 {{nowrap|1 − ''s''}} and ''s'' having the same information content. Hence it is the zeta function between {{nowrap|''s'' {{=}} {{sfrac|1|2}}}} and {{nowrap|''s'' {{=}} 1}}, and especially the zeta function along the critical line {{nowrap|''s'' {{=}} {{sfrac|1|2}}}}, which is of the most interest.

~~==== Introduction to Gram points ====~~

As {{nowrap|''s'' > 1}} gets larger, the Dirichlet series for the zeta function is increasingly dominated by the 2 term, getting ever closer to simply {{nowrap|1 + 2−''z''}}, which approaches 1 as {{nowrap|''s'' {{=}} Re(''z'')}} becomes larger. When {{nowrap|''s'' ≫ 1}} and ''x'' is an integer, the real part of zeta is approximately {{nowrap|1 + 2−''s''}}, and the imaginary part is approximately zero; that is, zeta is approximately real. Starting from {{nowrap|''s'' {{=}} +∞}} with ''x'' an integer, we can trace a line back towards the critical strip on which zeta is real. Since when {{nowrap|''s'' ≫ 1}} the derivative is approximately −{{sfrac|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 {{nowrap|''s'' {{=}} {{sfrac|1|2}}}}, it produces a real value of zeta on the critical line. Points on the critical line where {{nowrap|ζ({{frac|1|2}} + ''ig'')}} are real are called "Gram points", after {{w|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.

~~==== Gram points and zeta peaks ====~~

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 {{w|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 {{w|Riemann–Siegel formula}} since {{w|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 {{nowrap|ζ({{frac|1|2}} + ''ig'')}} at the corresponding Gram point should be especially large.

~~=== The Z function: a mathematically convenient version of zeta ===~~

The absolute value of {{nowrap|ζ({{frac|1|2}} + ''ig'')}} 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 {{nowrap|ζ({{frac|1|2}} + ''it'')}} with respect to ''t'' will be zero; however this does not mean its derivative there will be zero. In fact, the {{w|Riemann hypothesis}} is equivalent to the claim that all zeros of {{nowrap|ζ′(''s'' + ''it'')}} occur when {{nowrap|''s'' > {{sfrac|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 {{w|''Z'' function}}, which is defined (in terms of the [[The Riemann zeta function and tuning/Appendix#3. Z function and Riemann-Siegel theta function|Riemann-Siegel theta function]]) as:

~~<math>Z(t) = \exp(i \theta(t)) \zeta\left(\frac{1}{2} + it\right)</math>.~~

The factor of <math>\exp(i \theta(t))</math> simply modifies zeta by a complex phase, and so the absolute value of Z along the real axis is the same as the absolute value of ζ at the corresponding place on the critical line, and the zeros of Z in this strip correspond one to one with the zeros of ζ in the critical strip, and since θ is holomorphic on the strip with imaginary part between −{{sfrac|1|2}} and {{sfrac|1|2}}, so is Z. And Z is a real even function of the real variable ''t'', since theta was defined so as to give precisely this property.

~~=== Plots ===~~

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 {{nowrap|''t'' {{=}} {{sfrac|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 {{w|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 Mathematica-generated plot of Z({{frac|2π''x''|ln(2)}}) in the region around 12edo:

~~[[File:plot12.png|alt=plot12.png|plot12.png]]~~

The peak around 12 is both higher and wider than the local maximums above 11 and 13, indicating its superiority as an edo. Note also that the peak occurs at a point slightly larger than 12; this indicates the octave is slightly compressed in the zeta tuning for 12. The size of a step in octaves is 1/''x'', and hence the size of the octave in the zeta peak value tuning for ''N''edo is ''N''/''x''; if ''x'' is slightly larger than ''N'' as here with {{nowrap|''N'' {{=}} 12}}, the size of the zeta tuned octave will be slightly less than a pure octave. Similarly, when the peak occurs with ''x'' less than ''N'', we have stretched octaves.

For larger edos, the width of the peak narrows, but for strong edos the height more than compensates, measured in terms of the area under the peak (the absolute value of the integral of Z between two zeros.) Note how 270 completely dominates its neighbors:

~~[[File:plot270.png|alt=plot270.png|plot270.png]]~~

Note that for one of its neighbors, 271, it isn't entirely clear which peak value corresponds to the line of real values from +∞. This can be determined by looking at the absolute value of zeta along other ''s'' values, such as {{nowrap|''s'' {{=}} 1}} or {{nowrap|''s'' {{=}} {{sfrac|3|4}}}}, and in this case the local minimum at 271.069 is the value in question. However, other peak values are not without their interest; the local maximum at 270.941, for instance, is associated to a different mapping for 3.

To generate this plot using the free version of Wolfram Cloud, you can run <code>Plot[Abs[RiemannSiegelZ[9.06472028x]], {x, 11.9, 12.1}]</code> and then in the menu select '''Evaluation > Evaluate Cells'''. Change "'''11.9'''" and "'''12.1'''" to whatever values you want, e.g. to view the curve around 15edo you might use the values "'''14.9'''" and "'''15.1'''".

~~You can also view the plot using [https://www.desmos.com/calculator/dstp7wnidf Zeta in Desmos].~~

== Mike Battaglia's expanded results ==

Line 336:

Line 320:

More can be found at the page on [[Harmonic entropy#Extending_HE_to_.5Bmath.5DN.3D.5Cinfty.5B.2Fmath.5D:_zeta-HE|harmonic entropy]], including a generalization to Renyi entropy for arbitrary <math>a</math>.

== The matter of sigma: the critical strip, zeta peaks, and Gram points ==

So long as {{nowrap|''s'' ≥ 1}}, 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 {{nowrap|''s'' < 1}}, particularly in the [http://mathworld.wolfram.com/CriticalStrip.html critical strip], when {{nowrap|0 < ''s'' < 1}}. As s approaches the value {{nowrap|''s'' {{=}} {{sfrac|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 [https://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 {{nowrap|1 − ''s''}} and ''s'' having the same information content. Hence it is the zeta function between {{nowrap|''s'' {{=}} {{sfrac|1|2}}}} and {{nowrap|''s'' {{=}} 1}}, and especially the zeta function along the critical line {{nowrap|''s'' {{=}} {{sfrac|1|2}}}}, which is of the most interest.

=== Introduction to Gram points ===

As {{nowrap|''s'' > 1}} gets larger, the Dirichlet series for the zeta function is increasingly dominated by the 2 term, getting ever closer to simply {{nowrap|1 + 2−''z''}}, which approaches 1 as {{nowrap|''s'' {{=}} Re(''z'')}} becomes larger. When {{nowrap|''s'' ≫ 1}} and ''x'' is an integer, the real part of zeta is approximately {{nowrap|1 + 2−''s''}}, and the imaginary part is approximately zero; that is, zeta is approximately real. Starting from {{nowrap|''s'' {{=}} +∞}} with ''x'' an integer, we can trace a line back towards the critical strip on which zeta is real. Since when {{nowrap|''s'' ≫ 1}} the derivative is approximately −{{sfrac|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 {{nowrap|''s'' {{=}} {{sfrac|1|2}}}}, it produces a real value of zeta on the critical line. Points on the critical line where {{nowrap|ζ({{frac|1|2}} + ''ig'')}} are real are called "Gram points", after {{w|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.

=== Gram points and zeta peaks ===

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 {{w|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 {{w|Riemann–Siegel formula}} since {{w|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 {{nowrap|ζ({{frac|1|2}} + ''ig'')}} at the corresponding Gram point should be especially large.

=== The Z function: a mathematically convenient version of zeta ===

The absolute value of {{nowrap|ζ({{frac|1|2}} + ''ig'')}} 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 {{nowrap|ζ({{frac|1|2}} + ''it'')}} with respect to ''t'' will be zero; however this does not mean its derivative there will be zero. In fact, the {{w|Riemann hypothesis}} is equivalent to the claim that all zeros of {{nowrap|ζ′(''s'' + ''it'')}} occur when {{nowrap|''s'' > {{sfrac|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 {{w|''Z'' function}}, which is defined (in terms of the [[The Riemann zeta function and tuning/Appendix#3. Z function and Riemann-Siegel theta function|Riemann-Siegel theta function]]) as:

<math>Z(t) = \exp(i \theta(t)) \zeta\left(\frac{1}{2} + it\right)</math>.

The factor of <math>\exp(i \theta(t))</math> simply modifies zeta by a complex phase, and so the absolute value of Z along the real axis is the same as the absolute value of ζ at the corresponding place on the critical line, and the zeros of Z in this strip correspond one to one with the zeros of ζ in the critical strip, and since θ is holomorphic on the strip with imaginary part between −{{sfrac|1|2}} and {{sfrac|1|2}}, so is Z. And Z is a real even function of the real variable ''t'', since theta was defined so as to give precisely this property.

== Zeta edo lists ==

@@ Line 41: / Line 41: @@
 You may also view the graph of zeta along the critical line on Desmos: [https://www.desmos.com/calculator/dstp7wnidf Zeta in Desmos]. This makes it easier to see peaks, but only works for sigma=1/2.
+=== Plots ===
+Below are some demonstrative plots of the zeta function (strictly speaking, the [[#The Z function: a mathematically convenient version of zeta|Z function]]) on the critical line.
+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 {{nowrap|''t'' {{=}} {{sfrac|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 {{w|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 Mathematica-generated plot of Z({{frac|2π''x''|ln(2)}}) in the region around 12edo:
+[[File:plot12.png|alt=plot12.png|plot12.png]]
+The peak around 12 is both higher and wider than the local maximums above 11 and 13, indicating its superiority as an edo. Note also that the peak occurs at a point slightly larger than 12; this indicates the octave is slightly compressed in the zeta tuning for 12. The size of a step in octaves is 1/''x'', and hence the size of the octave in the zeta peak value tuning for ''N''edo is ''N''/''x''; if ''x'' is slightly larger than ''N'' as here with {{nowrap|''N'' {{=}} 12}}, the size of the zeta tuned octave will be slightly less than a pure octave. Similarly, when the peak occurs with ''x'' less than ''N'', we have stretched octaves.
+For larger edos, the width of the peak narrows, but for strong edos the height more than compensates, measured in terms of the area under the peak (the absolute value of the integral of Z between two zeros.) Note how 270 completely dominates its neighbors:
+[[File:plot270.png|alt=plot270.png|plot270.png]]
+Note that for one of its neighbors, 271, it isn't entirely clear which peak value corresponds to the line of real values from +∞. This can be determined by looking at the absolute value of zeta along other ''s'' values, such as {{nowrap|''s'' {{=}} 1}} or {{nowrap|''s'' {{=}} {{sfrac|3|4}}}}, and in this case the local minimum at 271.069 is the value in question. However, other peak values are not without their interest; the local maximum at 270.941, for instance, is associated to a different mapping for 3.
+To generate this plot using the free version of Wolfram Cloud, you can run <code>Plot[Abs[RiemannSiegelZ[9.06472028x]], {x, 11.9, 12.1}]</code> and then in the menu select '''Evaluation &gt; Evaluate Cells'''. Change "'''11.9'''" and "'''12.1'''" to whatever values you want, e.g. to view the curve around 15edo you might use the values "'''14.9'''" and "'''15.1'''".
+You can also view the plot using [https://www.desmos.com/calculator/dstp7wnidf Zeta in Desmos].
 == Gene Smith's original derivation ==
-=== The relation of zeta to tuning ===
 Suppose ''x'' is a variable representing some equal division of the octave. For example, if {{nowrap|''x'' {{=}} 80}}, ''x'' reflects 80edo with a step size of 15 cents and with pure octaves. Suppose that ''x'' can also be continuous, so that it can also represent fractional or "nonoctave" divisions as well. The [[Bohlen–Pierce scale]], 13 equal divisions of 3/1, is approximately 8.202 equal divisions of the "octave" (although the octave itself does not appear in this tuning), and would hence be represented by a value of {{nowrap|''x'' {{=}} 8.202}}.
@@ Line 116: / Line 136: @@
 so that we see that the absolute value of the zeta function serves to measure the relative error of an equal division.
-=== The critical strip, zeta peaks, and Gram points ===
-So long as {{nowrap|''s'' &ge; 1}}, 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 {{nowrap|''s'' &lt; 1}}, particularly in the [http://mathworld.wolfram.com/CriticalStrip.html critical strip], when {{nowrap|0 &lt; ''s'' &lt; 1}}. As s approaches the value {{nowrap|''s'' {{=}} {{sfrac|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 [https://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 {{nowrap|1 − ''s''}} and ''s'' having the same information content. Hence it is the zeta function between {{nowrap|''s'' {{=}} {{sfrac|1|2}}}} and {{nowrap|''s'' {{=}} 1}}, and especially the zeta function along the critical line {{nowrap|''s'' {{=}} {{sfrac|1|2}}}}, which is of the most interest.
-==== Introduction to Gram points ====
-As {{nowrap|''s'' &gt; 1}} gets larger, the Dirichlet series for the zeta function is increasingly dominated by the 2 term, getting ever closer to simply {{nowrap|1 + 2<sup>−''z''</sup>}}, which approaches 1 as {{nowrap|''s'' {{=}} Re(''z'')}} becomes larger. When {{nowrap|''s'' ≫ 1}} and ''x'' is an integer, the real part of zeta is approximately {{nowrap|1 + 2<sup>−''s''</sup>}}, and the imaginary part is approximately zero; that is, zeta is approximately real. Starting from {{nowrap|''s'' {{=}} +&infin;}} with ''x'' an integer, we can trace a line back towards the critical strip on which zeta is real. Since when {{nowrap|''s'' ≫ 1}} the derivative is approximately −{{sfrac|ln(2)|2<sup>''s''</sup>}}, 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 {{nowrap|''s'' {{=}} {{sfrac|1|2}}}}, it produces a real value of zeta on the critical line. Points on the critical line where {{nowrap|ζ({{frac|1|2}} + ''ig'')}} are real are called "Gram points", after {{w|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.
-==== Gram points and zeta peaks ====
-Because the value of zeta increased continuously as it made its way from +&infin; 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 {{w|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 {{w|Riemann–Siegel formula}} since {{w|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 {{nowrap|ζ({{frac|1|2}} + ''ig'')}} at the corresponding Gram point should be especially large.
-=== The Z function: a mathematically convenient version of zeta ===
-The absolute value of {{nowrap|ζ({{frac|1|2}} + ''ig'')}} 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 {{nowrap|ζ({{frac|1|2}} + ''it'')}} with respect to ''t'' will be zero; however this does not mean its derivative there will be zero. In fact, the {{w|Riemann hypothesis}} is equivalent to the claim that all zeros of {{nowrap|ζ′(''s'' + ''it'')}} occur when {{nowrap|''s'' &gt; {{sfrac|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 {{w|''Z'' function}}, which is defined (in terms of the [[The Riemann zeta function and tuning/Appendix#3. Z function and Riemann-Siegel theta function|Riemann-Siegel theta function]]) as:
-<math>Z(t) = \exp(i \theta(t)) \zeta\left(\frac{1}{2} + it\right)</math>.
-The factor of <math>\exp(i \theta(t))</math> simply modifies zeta by a complex phase, and so the absolute value of Z along the real axis is the same as the absolute value of ζ at the corresponding place on the critical line, and the zeros of Z in this strip correspond one to one with the zeros of ζ in the critical strip, and since θ is holomorphic on the strip with imaginary part between −{{sfrac|1|2}} and {{sfrac|1|2}}, so is Z. And Z is a real even function of the real variable ''t'', since theta was defined so as to give precisely this property.
-=== Plots ===
-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 {{nowrap|''t'' {{=}} {{sfrac|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 {{w|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 Mathematica-generated plot of Z({{frac|2π''x''|ln(2)}}) in the region around 12edo:
-[[File:plot12.png|alt=plot12.png|plot12.png]]
-The peak around 12 is both higher and wider than the local maximums above 11 and 13, indicating its superiority as an edo. Note also that the peak occurs at a point slightly larger than 12; this indicates the octave is slightly compressed in the zeta tuning for 12. The size of a step in octaves is 1/''x'', and hence the size of the octave in the zeta peak value tuning for ''N''edo is ''N''/''x''; if ''x'' is slightly larger than ''N'' as here with {{nowrap|''N'' {{=}} 12}}, the size of the zeta tuned octave will be slightly less than a pure octave. Similarly, when the peak occurs with ''x'' less than ''N'', we have stretched octaves.
-For larger edos, the width of the peak narrows, but for strong edos the height more than compensates, measured in terms of the area under the peak (the absolute value of the integral of Z between two zeros.) Note how 270 completely dominates its neighbors:
-[[File:plot270.png|alt=plot270.png|plot270.png]]
-Note that for one of its neighbors, 271, it isn't entirely clear which peak value corresponds to the line of real values from +∞. This can be determined by looking at the absolute value of zeta along other ''s'' values, such as {{nowrap|''s'' {{=}} 1}} or {{nowrap|''s'' {{=}} {{sfrac|3|4}}}}, and in this case the local minimum at 271.069 is the value in question. However, other peak values are not without their interest; the local maximum at 270.941, for instance, is associated to a different mapping for 3.
-To generate this plot using the free version of Wolfram Cloud, you can run <code>Plot[Abs[RiemannSiegelZ[9.06472028x]], {x, 11.9, 12.1}]</code> and then in the menu select '''Evaluation &gt; Evaluate Cells'''. Change "'''11.9'''" and "'''12.1'''" to whatever values you want, e.g. to view the curve around 15edo you might use the values "'''14.9'''" and "'''15.1'''".
-You can also view the plot using [https://www.desmos.com/calculator/dstp7wnidf Zeta in Desmos].
 == Mike Battaglia's expanded results ==
@@ Line 336: / Line 320: @@
 More can be found at the page on [[Harmonic entropy#Extending_HE_to_.5Bmath.5DN.3D.5Cinfty.5B.2Fmath.5D:_zeta-HE|harmonic entropy]], including a generalization to Renyi entropy for arbitrary <math>a</math>.
+== The matter of sigma: the critical strip, zeta peaks, and Gram points ==
+So long as {{nowrap|''s'' &ge; 1}}, 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 {{nowrap|''s'' &lt; 1}}, particularly in the [http://mathworld.wolfram.com/CriticalStrip.html critical strip], when {{nowrap|0 &lt; ''s'' &lt; 1}}. As s approaches the value {{nowrap|''s'' {{=}} {{sfrac|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 [https://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 {{nowrap|1 − ''s''}} and ''s'' having the same information content. Hence it is the zeta function between {{nowrap|''s'' {{=}} {{sfrac|1|2}}}} and {{nowrap|''s'' {{=}} 1}}, and especially the zeta function along the critical line {{nowrap|''s'' {{=}} {{sfrac|1|2}}}}, which is of the most interest.
+=== Introduction to Gram points ===
+As {{nowrap|''s'' &gt; 1}} gets larger, the Dirichlet series for the zeta function is increasingly dominated by the 2 term, getting ever closer to simply {{nowrap|1 + 2<sup>−''z''</sup>}}, which approaches 1 as {{nowrap|''s'' {{=}} Re(''z'')}} becomes larger. When {{nowrap|''s'' ≫ 1}} and ''x'' is an integer, the real part of zeta is approximately {{nowrap|1 + 2<sup>−''s''</sup>}}, and the imaginary part is approximately zero; that is, zeta is approximately real. Starting from {{nowrap|''s'' {{=}} +&infin;}} with ''x'' an integer, we can trace a line back towards the critical strip on which zeta is real. Since when {{nowrap|''s'' ≫ 1}} the derivative is approximately −{{sfrac|ln(2)|2<sup>''s''</sup>}}, 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 {{nowrap|''s'' {{=}} {{sfrac|1|2}}}}, it produces a real value of zeta on the critical line. Points on the critical line where {{nowrap|ζ({{frac|1|2}} + ''ig'')}} are real are called "Gram points", after {{w|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.
+=== Gram points and zeta peaks ===
+Because the value of zeta increased continuously as it made its way from +&infin; 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 {{w|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 {{w|Riemann–Siegel formula}} since {{w|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 {{nowrap|ζ({{frac|1|2}} + ''ig'')}} at the corresponding Gram point should be especially large.
+=== The Z function: a mathematically convenient version of zeta ===
+The absolute value of {{nowrap|ζ({{frac|1|2}} + ''ig'')}} 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 {{nowrap|ζ({{frac|1|2}} + ''it'')}} with respect to ''t'' will be zero; however this does not mean its derivative there will be zero. In fact, the {{w|Riemann hypothesis}} is equivalent to the claim that all zeros of {{nowrap|ζ′(''s'' + ''it'')}} occur when {{nowrap|''s'' &gt; {{sfrac|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 {{w|''Z'' function}}, which is defined (in terms of the [[The Riemann zeta function and tuning/Appendix#3. Z function and Riemann-Siegel theta function|Riemann-Siegel theta function]]) as:
+<math>Z(t) = \exp(i \theta(t)) \zeta\left(\frac{1}{2} + it\right)</math>.
+The factor of <math>\exp(i \theta(t))</math> simply modifies zeta by a complex phase, and so the absolute value of Z along the real axis is the same as the absolute value of ζ at the corresponding place on the critical line, and the zeros of Z in this strip correspond one to one with the zeros of ζ in the critical strip, and since θ is holomorphic on the strip with imaginary part between −{{sfrac|1|2}} and {{sfrac|1|2}}, so is Z. And Z is a real even function of the real variable ''t'', since theta was defined so as to give precisely this property.
 == Zeta edo lists ==