The Riemann zeta function and tuning: Difference between revisions

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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.

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 {{nowrap|1 − cos(2π''x'')}}, which is a smooth and in fact an [[Wikipedia:entire function|entire function]]. Let us therefore now define for any {{nowrap|''s'' > 1}}:

<math>\displaystyle E_s(x) = \sum_{n \geq 1} \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 real [[Wikipedia:analytic function|analytic function]] defined for all ''x'', and hence with all the smoothness properties we could desire.

For any fixed {{nowrap|''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:

Line 49:

=== Into the critical strip ===

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'' {{=}} {{frac|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'' {{=}} {{frac|1|2}}}} and {{nowrap|''s'' {{=}} 1}}, and especially the zeta function along the critical line {{nowrap|''s'' {{=}} {{frac|1|2}}}}, which is of the most interest.

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 −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'' {{=}} {{frac|2}}}}, it produces a real value of zeta on the critical line. Points on the critical line where {{nowrap|ζ({{frac|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.

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 {{nowrap|−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'' {{=}} {{frac|2}}}}, it produces a real value of zeta on the critical line. Points on the critical line where {{nowrap|ζ({{frac|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 [[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 {{nowrap|ζ({{frac|2}} + i''g'')}} at the corresponding Gram point should be especially large.

=== The Z function ===

The absolute value of {{nowrap|ζ({{frac|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 {{nowrap|ζ({{frac|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 {{nowrap|ζ'(''s'' + i''t'')}} occur when {{nowrap|''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]].

The absolute value of {{nowrap|ζ({{frac|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 {{nowrap|ζ({{frac|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 {{nowrap|ζ'(''s'' + i''t'')}} occur when {{nowrap|''s'' > {{frac|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 [[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

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<math>\displaystyle \Upsilon(z) = -\gamma z - \ln z + \sum_{k=1}^\infty \frac{z}{k} - \ln\left(1 + \frac{z}{k}\right)</math>

where ~~the lower-case~~ gamma is the [[Wikipedia:Euler–Mascheroni constant|~~Euler–Mascheroni~~ constant]]. We now may define the Riemann-Siegel theta function as

where &gamma is the [[Wikipedia:Euler–Mascheroni constant|Euler–Mascheroni 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>

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- \arctan\left(\frac{2t}{4n+1}\right)\right)</math>

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 θ 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 {{nowrap|{{!}}''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 θ is holomorphic in the strip with imaginary part between −{{frac|1|2}} and {{frac|1|2}}. It may be described for real arguments as an odd real analytic function of ''x'', increasing when {{nowrap|{{!}}''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 [[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\left(\frac{1}{2} + it\right)</math>

Since θ is holomorphic on the strip with imaginary part between -1/2 and 1/2, so is Z. Since the exponential function has no zeros, the zeros of Z in this strip correspond one to one with the zeros of ζ in the critical strip. Since the exponential of an imaginary argument has absolute value 1, 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 since theta was defined so as to give precisely this property, Z is a real even function of the real variable ''t''.

Since θ is holomorphic on the strip with imaginary part between −{{frac|1|2}} and {{frac|1|2}}, so is Z. Since the exponential function has no zeros, the zeros of Z in this strip correspond one to one with the zeros of ζ in the critical strip. Since the exponential of an imaginary argument has absolute value 1, 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 since theta was defined so as to give precisely this property, Z is a real even function of the real variable ''t''.

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

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 Mathematica-generated plot of Z({{nowrap|2π''x'' / ln(2)}}) in the region around 12edo:

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

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[[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'' {{=}} 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.

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'' {{=}} {{frac|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 copy-paste ~~'''~~Plot[Abs[RiemannSiegelZ[9.06472028x]], {x, 11.9,12.1}]~~'''~~ 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'''".

To generate this plot using the free version of Wolfram Cloud, you can copy-paste <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].

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where the product is over all primes p. The product converges for values of s with real part greater than or equal to one, except for s=1 where it diverges to infinity. We may remove a finite list of primes from consideration by multiplying ζ(s) by the corresponding factors {{nowrap|(1 − p−s)}} for each prime p we wish to remove. After we have done this, the smallest prime remaining will dominate peak values for s with large real part, and as before we can track these peaks backwards and, by analytical continuation, into the critical strip. In particular if we remove the prime 2, {{nowrap|(1 − 2−s)ζ(s)}} is now dominated by 3, and the large peak values occur near equal divisions of the "tritave", ie 3.

Along the critical line, {{nowrap|{{!}}1 − p−1/2 − it{{!}}}} may be written

Along the critical line, {{nowrap|{{!}}1 − p−{{frac|1|2}} − it{{!}}}} may be written

<math>\displaystyle{

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=== 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 {{nowrap|ζ(1/2 + ig)}} is real. This implies that exp(iθ(g)) is real, so that {{frac|θ(g)|π}} is an integer. Theta has an [[Wikipedia:asymptotic expansion|asymptotic expansion]]

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 {{nowrap|ζ({{frac|1|2}} + ig)}} is real. This implies that exp(iθ(g)) is real, so that {{frac|θ(g)|π}} is an integer. Theta has an [[Wikipedia:asymptotic expansion|asymptotic expansion]]

<math>\displaystyle{

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* [http://terrytao.wordpress.com/2009/07/12/selbergs-limit-theorem-for-the-riemann-zeta-function-on-the-critical-line/ Selberg's limit theorem] by Terence Tao [http://www.webcitation.org/5xrvgjW6T Permalink]

* [[:File:Zetamusic5.pdf|Favored cardinalities of scales]] by Peter Buch

* [http://www.ams.org/journals/mcom/2004-73-246/S0025-5718-03-01568-0/S0025-5718-03-01568-0.pdf Computational estimation of the order of {{nowrap|ζ(1/2 + it)}}] by Tadej Kotnik

* [http://www.ams.org/journals/mcom/2004-73-246/S0025-5718-03-01568-0/S0025-5718-03-01568-0.pdf Computational estimation of the order of {{nowrap|ζ({{farc|1|2}} + it)}}] by Tadej Kotnik

* [https://www-users.cse.umn.edu/~odlyzko/zeta_tables/index.html Andrew Odlyzko: Tables of zeros of the Riemann zeta function]

* [https://www-users.cse.umn.edu/~odlyzko/doc/zeta.html Andrew Odlyzko: Papers on Zeros of the Riemann Zeta Function and Related Topics]

@@ Line 28: / Line 28: @@
 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 [[Wikipedia:Continuous_function#Relation_to_differentiability_and_integrability|discontinuous derivative]], whereas a smooth function be preferred. The function &#x230A;x&#x2309;<sup>2</sup> is quadratically increasing near integer values of x, and is periodic with period 1. Another function with these same properties is {{nowrap|1 &minus; cos(2π''x'')}}, which is a smooth and in fact an [[Wikipedia:entire function|entire function]]. Let us therefore now define for any {{nowrap|''s'' > 1}}:
+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 &#x230A;x&#x2309;<sup>2</sup> is quadratically increasing near integer values of x, and is periodic with period 1. Another function with these same properties is {{nowrap|1 &minus; cos(2π''x'')}}, which is a smooth and in fact an [[Wikipedia:entire function|entire function]]. Let us therefore now define for any {{nowrap|''s'' &gt; 1}}:
 <math>\displaystyle E_s(x) = \sum_{n \geq 1} \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 real [[Wikipedia:analytic function|analytic function]] defined for all ''x'', and hence with all the smoothness properties we could desire.
+For any fixed {{nowrap|''s'' &gt; 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:
@@ Line 49: / Line 49: @@
 === Into the critical strip ===
-So long as {{nowrap|''s'' &#x2264; 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'' {{=}} {{frac|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 &minus; ''s''}} and ''s'' having the same information content. Hence it is the zeta function between {{nowrap|''s'' {{=}} {{frac|1|2}}}} and {{nowrap|''s'' {{=}} 1}}, and especially the zeta function along the critical line {{nowrap|''s'' {{=}} {{frac|1|2}}}}, which is of the most interest.
+So long as {{nowrap|''s'' &#x2264; 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'' {{=}} {{frac|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 &minus; ''s''}} and ''s'' having the same information content. Hence it is the zeta function between {{nowrap|''s'' {{=}} {{frac|1|2}}}} and {{nowrap|''s'' {{=}} 1}}, and especially the zeta function along the critical line {{nowrap|''s'' {{=}} {{frac|1|2}}}}, which is of the most interest.
-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<sup>&minus;''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>&minus;''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 &minus;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'' {{=}} {{frac|2}}}}, it produces a real value of zeta on the critical line. Points on the critical line where {{nowrap|&zeta;({{frac|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.
+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<sup>&minus;''z''</sup>}}, which approaches 1 as {{nowrap|''s'' {{=}} Re(''z'')}} becomes larger. When {{nowrap|''s'' &gt;&gt; 1}} and ''x'' is an integer, the real part of zeta is approximately {{nowrap|1 + 2<sup>&minus;''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'' &gt;&gt; 1}} the derivative is approximately {{nowrap|&minus;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'' {{=}} {{frac|2}}}}, it produces a real value of zeta on the critical line. Points on the critical line where {{nowrap|&zeta;({{frac|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 +&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 &minus;&zeta;'(''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 {{nowrap|&zeta;({{frac|2}} + i''g'')}} at the corresponding Gram point should be especially large.
 === The Z function ===
-The absolute value of {{nowrap|&zeta;({{frac|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 {{nowrap|&zeta;({{frac|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 {{nowrap|&zeta;'(''s'' + i''t'')}} occur when {{nowrap|''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]].
+The absolute value of {{nowrap|&zeta;({{frac|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 {{nowrap|&zeta;({{frac|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 {{nowrap|&zeta;'(''s'' + i''t'')}} occur when {{nowrap|''s'' &gt; {{frac|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 [[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
@@ Line 62: / Line 62: @@
 <math>\displaystyle \Upsilon(z) = -\gamma z - \ln z + \sum_{k=1}^\infty \frac{z}{k} - \ln\left(1 + \frac{z}{k}\right)</math>
-where the lower-case gamma is the [[Wikipedia:Euler–Mascheroni constant|Euler–Mascheroni constant]]. We now may define the Riemann-Siegel theta function as
+where &gamma is the [[Wikipedia:Euler–Mascheroni constant|Euler&ndash;Mascheroni 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>
@@ Line 72: / Line 72: @@
 - \arctan\left(\frac{2t}{4n+1}\right)\right)</math>
-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 {{nowrap|{{!}}''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 &minus;{{frac|1|2}} and {{frac|1|2}}. It may be described for real arguments as an odd real analytic function of ''x'', increasing when {{nowrap|{{!}}''x''{{!}} &gt; 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 [[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\left(\frac{1}{2} + it\right)</math>
-Since &theta; is holomorphic on the strip with imaginary part between -1/2 and 1/2, so is Z. Since the exponential function has no zeros, the zeros of Z in this strip correspond one to one with the zeros of &zeta; in the critical strip. Since the exponential of an imaginary argument has absolute value 1, the absolute value of Z along the real axis is the same as the absolute value of &zeta; at the corresponding place on the critical line. And since theta was defined so as to give precisely this property, Z is a real even function of the real variable ''t''.
+Since &theta; is holomorphic on the strip with imaginary part between &minus;{{frac|1|2}} and {{frac|1|2}}, so is Z. Since the exponential function has no zeros, the zeros of Z in this strip correspond one to one with the zeros of &zeta; in the critical strip. Since the exponential of an imaginary argument has absolute value 1, the absolute value of Z along the real axis is the same as the absolute value of &zeta; at the corresponding place on the critical line. And since theta was defined so as to give precisely this property, Z is a real even function of the real variable ''t''.
-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'' {{=}} 2π''x''/ln(2)}}, 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 {{nowrap|''t'' {{=}} 2π''x'' / ln(2)}}, 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 [[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 Mathematica-generated plot of Z(2π''x''/ln(2)) in the region around 12edo:
+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 Mathematica-generated plot of Z({{nowrap|2π''x'' / ln(2)}}) in the region around 12edo:
 [[File:plot12.png|alt=plot12.png|plot12.png]]
@@ Line 92: / Line 92: @@
 [[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'' {{=}} 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.
+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'' {{=}} {{frac|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 copy-paste '''Plot[Abs[RiemannSiegelZ[9.06472028x]], {x, 11.9,12.1}]''' 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'''".
+To generate this plot using the free version of Wolfram Cloud, you can copy-paste <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].
@@ Line 429: / Line 429: @@
 where the product is over all primes p. The product converges for values of s with real part greater than or equal to one, except for s=1 where it diverges to infinity. We may remove a finite list of primes from consideration by multiplying &zeta;(s) by the corresponding factors {{nowrap|(1 &minus; p<sup>&minus;s</sup>)}} for each prime p we wish to remove. After we have done this, the smallest prime remaining will dominate peak values for s with large real part, and as before we can track these peaks backwards and, by analytical continuation, into the critical strip. In particular if we remove the prime 2, {{nowrap|(1 &minus; 2<sup>&minus;s</sup>)&zeta;(s)}} is now dominated by 3, and the large peak values occur near equal divisions of the "tritave", ie 3.
-Along the critical line, {{nowrap|{{!}}1 &minus; p<sup>&minus;1/2 &minus; it</sup>{{!}}}} may be written
+Along the critical line, {{nowrap|{{!}}1 &minus; p<sup>&minus;{{frac|1|2}} &minus; it</sup>{{!}}}} may be written
 <math>\displaystyle{
@@ Line 440: / Line 440: @@
 === 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 &theta;(t). Recall that a Gram point is a point on the critical line where {{nowrap|&zeta;(1/2 + ig)}} is real. This implies that exp(i&theta;(g)) is real, so that {{frac|&theta;(g)|π}} is an integer. Theta has an [[Wikipedia:asymptotic expansion|asymptotic expansion]]
+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 &theta;(t). Recall that a Gram point is a point on the critical line where {{nowrap|&zeta;({{frac|1|2}} + ig)}} is real. This implies that exp(i&theta;(g)) is real, so that {{frac|&theta;(g)|π}} is an integer. Theta has an [[Wikipedia:asymptotic expansion|asymptotic expansion]]
 <math>\displaystyle{
@@ Line 471: / Line 471: @@
 * [http://terrytao.wordpress.com/2009/07/12/selbergs-limit-theorem-for-the-riemann-zeta-function-on-the-critical-line/ Selberg's limit theorem] by Terence Tao [http://www.webcitation.org/5xrvgjW6T Permalink]
 * [[:File:Zetamusic5.pdf|Favored cardinalities of scales]] by Peter Buch
-* [http://www.ams.org/journals/mcom/2004-73-246/S0025-5718-03-01568-0/S0025-5718-03-01568-0.pdf Computational estimation of the order of {{nowrap|&zeta;(1/2 + it)}}] by Tadej Kotnik
+* [http://www.ams.org/journals/mcom/2004-73-246/S0025-5718-03-01568-0/S0025-5718-03-01568-0.pdf Computational estimation of the order of {{nowrap|&zeta;({{farc|1|2}} + it)}}] by Tadej Kotnik
 * [https://www-users.cse.umn.edu/~odlyzko/zeta_tables/index.html Andrew Odlyzko: Tables of zeros of the Riemann zeta function]
 * [https://www-users.cse.umn.edu/~odlyzko/doc/zeta.html Andrew Odlyzko: Papers on Zeros of the Riemann Zeta Function and Related Topics]