The Riemann zeta function and tuning

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[[toc|flat]]
=Preliminaries= 
Suppose x is a continuous variable equal to the reciprocal of the step size of an equal division of the octave in fractions of an octave. For example, if the step size was 15 cents, then x = 1200/15 = 80, and we would be considering 80edo in pure octave tuning. If ||x|| denotes x minus x rounded to the nearest integer, or in other words the function x - floor(x+1/2), then the [[Tenney-Euclidean metrics|Tenney-Euclidean error]] for the [[p-limit]] [[val]] obtained by rounding log2(q) x to the nearest integer for each prime q up to p will be

[[math]]
 \sum_2^p (\frac{\|x \log_2 q\|}{\log_2 q})^2
[[math]] 

Now suppose we don't want a formula for any specific prime limit, but which applies to all primes. We can't take the above sum to infinity, since it doesn't converge. However, we could change the weighting factor to a power so that it does converge:

[[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 canceles 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 [[http://en.wikipedia.org/wiki/Von_Mangoldt_function|Von Mangoldt function]], an [[http://en.wikipedia.org/wiki/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, 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]]

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 [[http://en.wikipedia.org/wiki/Continuous_function|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(2 pi x), which is a smooth and in fact an [[http://en.wikipedia.org/wiki/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)}{n^s}
[[math]]

For any fixed s > 1 this gives a [[http://en.wikipedia.org/wiki/Analytic_function|real analytic function]] defined for all x, and hence with all the smoothness properties we could desire. We can define essentially the same function by subtracting it from E_s(1/2)/2:

[[math]]
F_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{\cos(2 \pi x)}{n^s}
[[math]]

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 [[http://en.wikipedia.org/wiki/Riemann_zeta_function|Riemann zeta function]]:

[[math]]
F_s(x) = \Re \ln \zeta(s + 2 \pi i x/\ln 2)
[[math]]

If we take exponentials of both sides, then

[[math]]
\exp(F_s(x)) = |\zeta(s + 2 \pi i x/\ln 2)|
[[math]]

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

=Into the critical strip=
So long as s is greater than or equal to one, the absolute value of the zeta function can be seen as an 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>0 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 +infinity 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 zeta(1/2 + i g) are real are called "Gram points", after [[http://en.wikipedia.org/wiki/J%C3%B8rgen_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 +infinity 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 -zeta'(z) is getting a boost from other small primes as it travels toward the Gram point. A complex formula due to [[http://en.wikipedia.org/wiki/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 [[http://en.wikipedia.org/wiki/Riemann%E2%80%93Siegel_formula|Riemann-Siegel formula]] since [[http://en.wikipedia.org/wiki/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 zeta(1/2 + i g) at the corresponding Gram point should be especially large.

=The Z function=
The absolute value zeta(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 zeta(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 [[http://en.wikipedia.org/wiki/Riemann_hypothesis|Riemann hypothesis]] is equivalent to the claim that all zeros of zeta'(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 [[http://en.wikipedia.org/wiki/Z_function|Z function]].

In order to define the Z function, we need first to define the [[http://en.wikipedia.org/wiki/Riemann%E2%80%93Siegel_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]]

where the lower-case gamma is [[http://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_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]]

Another approach is to substitute z = (1 + 2i t)/4 into the series for Log Gamma and take the imaginary part, this yields

[[math]]
\theta(t) = -\frac{\gamma + \log \pi}{2}t - \arctan 2t 
+ \sum_{n=1}^\infty \left(\frac{t}{2n} 
- \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 |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 [[http://en.wikipedia.org/wiki/Z_function|Z function]] as

[[math]]
Z(t) = \exp(i \theta(t)) \zeta(1/2 + it)
[[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.

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 = 2pi/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 [[http://en.wikipedia.org/wiki/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(2 pi x /ln(2)) in the region around 12edo:

[[image: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 Nedo is N/x; if x is slightly larger than N as here with 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:

[[image:plot12.png]]

=Computing zeta=
There are various approaches to the question of computing the zeta function, but perhaps the simplest is the use of the [[http://en.wikipedia.org/wiki/Dirichlet_eta_function|Dirichlet eta function]] which was introduced to mathematics by [[http://en.wikipedia.org/wiki/Johann_Peter_Gustav_Lejeune_Dirichlet|Johann Peter Gustav Lejeune Dirichlet]], who despite his name was a German and the brother-in-law of [[http://en.wikipedia.org/wiki/Felix_Mendelssohn_Bartholdy|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 [[http://en.wikipedia.org/wiki/Euler_product|Euler product]] or [[http://en.wikipedia.org/wiki/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}
= \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 + 2pi i x/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 [[http://en.wikipedia.org/wiki/Euler_summation|Euler summation]].

=Links=
[[http://front.math.ucdavis.edu/0309.5433|X-Ray of Riemann zeta-function]] by Juan Arias-de-Reyna
[[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]]

Original HTML content:

<html><head><title>The Riemann Zeta Function and Tuning</title></head><body><!-- ws:start:WikiTextTocRule:22:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --><a href="#Preliminaries">Preliminaries</a><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --> | <a href="#Into the critical strip">Into the critical strip</a><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --> | <a href="#The Z function">The Z function</a><!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --> | <a href="#Computing zeta">Computing zeta</a><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --> | <a href="#Links">Links</a><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: -->
<!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc0"><a name="Preliminaries"></a><!-- ws:end:WikiTextHeadingRule:12 -->Preliminaries</h1>
 Suppose x is a continuous variable equal to the reciprocal of the step size of an equal division of the octave in fractions of an octave. For example, if the step size was 15 cents, then x = 1200/15 = 80, and we would be considering 80edo in pure octave tuning. If ||x|| denotes x minus x rounded to the nearest integer, or in other words the function x - floor(x+1/2), then the <a class="wiki_link" href="/Tenney-Euclidean%20metrics">Tenney-Euclidean error</a> for the <a class="wiki_link" href="/p-limit">p-limit</a> <a class="wiki_link" href="/val">val</a> obtained by rounding log2(q) x to the nearest integer for each prime q up to p will be<br />
<br />
<!-- ws:start:WikiTextMathRule:0:
[[math]]&lt;br/&gt;
 \sum_2^p (\frac{\|x \log_2 q\|}{\log_2 q})^2&lt;br/&gt;[[math]]
 --><script type="math/tex"> \sum_2^p (\frac{\|x \log_2 q\|}{\log_2 q})^2</script><!-- ws:end:WikiTextMathRule:0 --> <br />
<br />
Now suppose we don't want a formula for any specific prime limit, but which applies to all primes. We can't take the above sum to infinity, since it doesn't converge. However, we could change the weighting factor to a power so that it does converge:<br />
<br />
<!-- ws:start:WikiTextMathRule:1:
[[math]]&lt;br/&gt;
 \sum_2^\infty \frac{\|x \log_2 q\|^2}{q^s}&lt;br/&gt;[[math]]
 --><script type="math/tex"> \sum_2^\infty \frac{\|x \log_2 q\|^2}{q^s}</script><!-- ws:end:WikiTextMathRule:1 --> <br />
<br />
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 canceles 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 <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Von_Mangoldt_function" rel="nofollow">Von Mangoldt function</a>, an <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Arithmetic_function" rel="nofollow">arithmetic function</a> 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, and in terms of it we can include prime powers in our error function as<br />
<br />
<!-- ws:start:WikiTextMathRule:2:
[[math]]&lt;br/&gt;
\sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{\|x \log_2 n\|^2}{n^s}&lt;br/&gt;[[math]]
 --><script type="math/tex">\sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{\|x \log_2 n\|^2}{n^s}</script><!-- ws:end:WikiTextMathRule:2 --><br />
<br />
where the summation is taken formally over all positive integers, though only the primes and prime powers make a nonzero contribution.<br />
<br />
Another consequence of the above definition which might be objected to is that it results in a function with a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Continuous_function" rel="nofollow">discontinuous derivative</a>, 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(2 pi x), which is a smooth and in fact an <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Entire_function" rel="nofollow">entire</a> function. Let us therefore now define for any s &gt; 1<br />
<br />
<!-- ws:start:WikiTextMathRule:3:
[[math]]&lt;br/&gt;
E_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{1 - \cos(2 \pi x)}{n^s}&lt;br/&gt;[[math]]
 --><script type="math/tex">E_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{1 - \cos(2 \pi x)}{n^s}</script><!-- ws:end:WikiTextMathRule:3 --><br />
<br />
For any fixed s &gt; 1 this gives a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Analytic_function" rel="nofollow">real analytic function</a> defined for all x, and hence with all the smoothness properties we could desire. We can define essentially the same function by subtracting it from E_s(1/2)/2:<br />
<br />
<!-- ws:start:WikiTextMathRule:4:
[[math]]&lt;br/&gt;
F_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{\cos(2 \pi x)}{n^s}&lt;br/&gt;[[math]]
 --><script type="math/tex">F_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{\cos(2 \pi x)}{n^s}</script><!-- ws:end:WikiTextMathRule:4 --><br />
<br />
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 <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Riemann_zeta_function" rel="nofollow">Riemann zeta function</a>:<br />
<br />
<!-- ws:start:WikiTextMathRule:5:
[[math]]&lt;br/&gt;
F_s(x) = \Re \ln \zeta(s + 2 \pi i x/\ln 2)&lt;br/&gt;[[math]]
 --><script type="math/tex">F_s(x) = \Re \ln \zeta(s + 2 \pi i x/\ln 2)</script><!-- ws:end:WikiTextMathRule:5 --><br />
<br />
If we take exponentials of both sides, then<br />
<br />
<!-- ws:start:WikiTextMathRule:6:
[[math]]&lt;br/&gt;
\exp(F_s(x)) = |\zeta(s + 2 \pi i x/\ln 2)|&lt;br/&gt;[[math]]
 --><script type="math/tex">\exp(F_s(x)) = |\zeta(s + 2 \pi i x/\ln 2)|</script><!-- ws:end:WikiTextMathRule:6 --><br />
<br />
so that we see that the absolute value of the zeta function serves to measure the error of an equal division.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc1"><a name="Into the critical strip"></a><!-- ws:end:WikiTextHeadingRule:14 -->Into the critical strip</h1>
So long as s is greater than or equal to one, the absolute value of the zeta function can be seen as an error measurement. However, the rationale for that view of things departs when s is less than one, particularly in the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/CriticalStrip.html" rel="nofollow">critical strip</a>, when s lies between zero and one. As s approaches the value s=1/2 of the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/CriticalLine.html" rel="nofollow">critical line</a>, 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 <a class="wiki_link_ext" href="http://planetmath.org/encyclopedia/FunctionalEquationOfTheRiemannZetaFunction.html" rel="nofollow">functional equation</a> 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.<br />
<br />
As s&gt;0 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 &gt;&gt; 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 +infinity with x an integer, we can trace a line back towards the critical strip on which zeta is real. Since when s &gt;&gt; 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 zeta(1/2 + i g) are real are called &quot;Gram points&quot;, after <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/J%C3%B8rgen_Pedersen_Gram" rel="nofollow">Jørgen Pedersen Gram</a>. 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.<br />
<br />
Because the value of zeta increased continuously as it made its way from +infinity 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 -zeta'(z) is getting a boost from other small primes as it travels toward the Gram point. A complex formula due to <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Riemann" rel="nofollow">Bernhard Riemann</a> which he failed to publish because it was so nasty becomes a bit simpler when used at a Gram point. It is named the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Riemann%E2%80%93Siegel_formula" rel="nofollow">Riemann-Siegel formula</a> since <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Carl_Ludwig_Siegel" rel="nofollow">Carl Ludwig Siegel</a> 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 zeta(1/2 + i g) at the corresponding Gram point should be especially large.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc2"><a name="The Z function"></a><!-- ws:end:WikiTextHeadingRule:16 -->The Z function</h1>
The absolute value zeta(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 zeta(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 <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Riemann_hypothesis" rel="nofollow">Riemann hypothesis</a> is equivalent to the claim that all zeros of zeta'(s + i t) occur when s &gt; 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 <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Z_function" rel="nofollow">Z function</a>.<br />
<br />
In order to define the Z function, we need first to define the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Riemann%E2%80%93Siegel_theta_function" rel="nofollow">Riemann-Siegel theta function</a>, and in order to do that, we first need to define the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/LogGammaFunction.html" rel="nofollow">Log Gamma function</a>. 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<br />
<br />
<!-- ws:start:WikiTextMathRule:7:
[[math]]&lt;br/&gt;
\Upsilon(z) = -\gamma z - \ln z + \sum_{k=1}^\infty \frac{z}{k} - \ln(1 + \frac{z}{k})&lt;br/&gt;[[math]]
 --><script type="math/tex">\Upsilon(z) = -\gamma z - \ln z + \sum_{k=1}^\infty \frac{z}{k} - \ln(1 + \frac{z}{k})</script><!-- ws:end:WikiTextMathRule:7 --><br />
<br />
where the lower-case gamma is <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant" rel="nofollow">Euler's constant</a>. We now may define the Riemann-Siegel theta function as<br />
<br />
<!-- ws:start:WikiTextMathRule:8:
[[math]]&lt;br/&gt;
\theta(z) = (\Upsilon((1 + 2 i z)/4) - \Upsilon((1 - 2 i z)/4))/(2 i) - \ln(\pi) z/2 &lt;br/&gt;[[math]]
 --><script type="math/tex">\theta(z) = (\Upsilon((1 + 2 i z)/4) - \Upsilon((1 - 2 i z)/4))/(2 i) - \ln(\pi) z/2 </script><!-- ws:end:WikiTextMathRule:8 --><br />
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Another approach is to substitute z = (1 + 2i t)/4 into the series for Log Gamma and take the imaginary part, this yields<br />
<br />
<!-- ws:start:WikiTextMathRule:9:
[[math]]&lt;br/&gt;
\theta(t) = -\frac{\gamma + \log \pi}{2}t - \arctan 2t &lt;br /&gt;
+ \sum_{n=1}^\infty \left(\frac{t}{2n} &lt;br /&gt;
- \arctan\left(\frac{2t}{4n+1}\right)\right)&lt;br/&gt;[[math]]
 --><script type="math/tex">\theta(t) = -\frac{\gamma + \log \pi}{2}t - \arctan 2t 
+ \sum_{n=1}^\infty \left(\frac{t}{2n} 
- \arctan\left(\frac{2t}{4n+1}\right)\right)</script><!-- ws:end:WikiTextMathRule:9 --><br />
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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| &gt; 6.29. Plots of it may be studied by use of the Wolfram <a class="wiki_link_ext" href="http://functions.wolfram.com/webMathematica/FunctionPlotting.jsp?name=RiemannSiegelTheta" rel="nofollow">online function plotter</a>.<br />
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Using the theta and zeta functions, we define the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Z_function" rel="nofollow">Z function</a> as<br />
<br />
<!-- ws:start:WikiTextMathRule:10:
[[math]]&lt;br/&gt;
Z(t) = \exp(i \theta(t)) \zeta(1/2 + it)&lt;br/&gt;[[math]]
 --><script type="math/tex">Z(t) = \exp(i \theta(t)) \zeta(1/2 + it)</script><!-- ws:end:WikiTextMathRule:10 --><br />
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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.<br />
<br />
Using the <a class="wiki_link_ext" href="http://functions.wolfram.com/webMathematica/FunctionPlotting.jsp?name=RiemannSiegelZ" rel="nofollow">online plotter</a> we can plot Z in the regions corresponding to scale divisions, using the conversion factor t = 2pi/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 <a class="wiki_link_ext" href="http://web.viu.ca/pughg/RiemannZeta/RiemannZetaLong.html" rel="nofollow">here</a>.<br />
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If you have access to <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Mathematica" rel="nofollow">Mathematica</a>, 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(2 pi x /ln(2)) in the region around 12edo:<br />
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<!-- ws:start:WikiTextLocalImageRule:29:&lt;img src=&quot;/file/view/plot12.png/219376858/plot12.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/plot12.png/219376858/plot12.png" alt="plot12.png" title="plot12.png" /><!-- ws:end:WikiTextLocalImageRule:29 --><br />
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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 Nedo is N/x; if x is slightly larger than N as here with 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.<br />
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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:<br />
<br />
<!-- ws:start:WikiTextLocalImageRule:30:&lt;img src=&quot;/file/view/plot12.png/219376858/plot12.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/plot12.png/219376858/plot12.png" alt="plot12.png" title="plot12.png" /><!-- ws:end:WikiTextLocalImageRule:30 --><br />
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<!-- ws:start:WikiTextHeadingRule:18:&lt;h1&gt; --><h1 id="toc3"><a name="Computing zeta"></a><!-- ws:end:WikiTextHeadingRule:18 -->Computing zeta</h1>
There are various approaches to the question of computing the zeta function, but perhaps the simplest is the use of the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Dirichlet_eta_function" rel="nofollow">Dirichlet eta function</a> which was introduced to mathematics by <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Johann_Peter_Gustav_Lejeune_Dirichlet" rel="nofollow">Johann Peter Gustav Lejeune Dirichlet</a>, who despite his name was a German and the brother-in-law of <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Felix_Mendelssohn_Bartholdy" rel="nofollow">Felix Mendelssohn</a>.<br />
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The zeta function has a <a class="wiki_link_ext" href="http://mathworld.wolfram.com/SimplePole.html" rel="nofollow">simple pole</a> at z=1 which forms a barrier against continuing it with its <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Euler_product" rel="nofollow">Euler product</a> or <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Dirichlet_series" rel="nofollow">Dirichlet series</a> 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:<br />
<br />
<!-- ws:start:WikiTextMathRule:11:
[[math]]&lt;br/&gt;
\eta(z) = (1-2^{1-z})\zeta(z) = \sum_{n=1}^\infty (-1)^{n-1} n^{-z}&lt;br /&gt;
= \frac{1}{1^z} - \frac{1}{2^z} + \frac{1}{3^z} - \frac{1}{4^z} + \cdots&lt;br/&gt;[[math]]
 --><script type="math/tex">\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</script><!-- ws:end:WikiTextMathRule:11 --><br />
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The Dirichlet series for the zeta function is absolutely convergent when s&gt;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 + 2pi i x/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 <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Euler_summation" rel="nofollow">Euler summation</a>.<br />
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<!-- ws:start:WikiTextHeadingRule:20:&lt;h1&gt; --><h1 id="toc4"><a name="Links"></a><!-- ws:end:WikiTextHeadingRule:20 -->Links</h1>
<a class="wiki_link_ext" href="http://front.math.ucdavis.edu/0309.5433" rel="nofollow">X-Ray of Riemann zeta-function</a> by Juan Arias-de-Reyna<br />
<a class="wiki_link_ext" href="http://terrytao.wordpress.com/2009/07/12/selbergs-limit-theorem-for-the-riemann-zeta-function-on-the-critical-line/" rel="nofollow">Selberg's limit theorem</a> by Terence Tao <a class="wiki_link_ext" href="http://www.webcitation.org/5xrvgjW6T" rel="nofollow">Permalink</a></body></html>