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The Riemann zeta function and tuning: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-04-10 14:47:33 UTC</tt>.<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-04-10 15:11:35 UTC</tt>.<br>
: The original revision id was <tt>218886326</tt>.<br>
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So long as s is greater than or equal to one, the absolute value of the zeta function can be seen as 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.
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.


The question now arises as to how we can continue the zeta function into the critical strip in a way which supports the connection with tuning and the claims above. There are various approaches to this, 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]].
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 approximatly zero; that is, zeta is 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, meaning that this real value for zeta increases as we go back. The zeta function approaches 1 uniformly as s increases to infinity, so going back, 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 sort of Gram point which corresponds to an edo.
 
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 [[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=
 
=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:
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:
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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 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]].
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 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]].
=Lines from infinity=
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 approximatly zero; that is, zeta is 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, meaning that this real value for zeta increases as we go back. The zeta function approaches 1 uniformly as s increases to infinity, so going back, 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 sort of Gram point which corresponds to an edo.
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 [[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.


=Links=
=Links=
[[http://front.math.ucdavis.edu/0309.5433|X-Ray of Riemann zeta-function]] by Juan Arias-de-Reyna</pre></div>
[[http://front.math.ucdavis.edu/0309.5433|X-Ray of Riemann zeta-function]] by Juan Arias-de-Reyna</pre></div>
<h4>Original HTML content:</h4>
<h4>Original HTML content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;The Riemann Zeta Function and Tuning&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:16:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt;&lt;a href="#Preliminaries"&gt;Preliminaries&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt; | &lt;a href="#Into the critical strip"&gt;Into the critical strip&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt; | &lt;a href="#Lines from infinity"&gt;Lines from infinity&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;!-- ws:start:WikiTextTocRule:20: --&gt; | &lt;a href="#Links"&gt;Links&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt;
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;The Riemann Zeta Function and Tuning&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:18:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt;&lt;a href="#Preliminaries"&gt;Preliminaries&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;!-- ws:start:WikiTextTocRule:20: --&gt; | &lt;a href="#Into the critical strip"&gt;Into the critical strip&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt; | &lt;a href="#The Z function"&gt;The Z function&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt; | &lt;a href="#Computing zeta"&gt;Computing zeta&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt; | &lt;a href="#Links"&gt;Links&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:23 --&gt;&lt;!-- ws:start:WikiTextTocRule:24: --&gt;
&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Preliminaries"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Preliminaries&lt;/h1&gt;
&lt;!-- ws:end:WikiTextTocRule:24 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Preliminaries"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Preliminaries&lt;/h1&gt;
  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 x - floor(x+1/2), then the &lt;a class="wiki_link" href="/Tenney-Euclidean%20metrics"&gt;Tenney-Euclidean error&lt;/a&gt; for the &lt;a class="wiki_link" href="/p-limit"&gt;p-limit&lt;/a&gt; &lt;a class="wiki_link" href="/val"&gt;val&lt;/a&gt; obtained by rounding log2(q) x to the nearest integer for each prime q up to p will be&lt;br /&gt;
  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 x - floor(x+1/2), then the &lt;a class="wiki_link" href="/Tenney-Euclidean%20metrics"&gt;Tenney-Euclidean error&lt;/a&gt; for the &lt;a class="wiki_link" href="/p-limit"&gt;p-limit&lt;/a&gt; &lt;a class="wiki_link" href="/val"&gt;val&lt;/a&gt; obtained by rounding log2(q) x to the nearest integer for each prime q up to p will be&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
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So long as s is greater than or equal to one, the absolute value of the zeta function can be seen as an error measurement. However, the rationale for that view of things departs when s is less than one, particularly in the &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/CriticalStrip.html" rel="nofollow"&gt;critical strip&lt;/a&gt;, when s lies between zero and one. As s approaches the value s=1/2 of the &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/CriticalLine.html" rel="nofollow"&gt;critical line&lt;/a&gt;, 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 &lt;a class="wiki_link_ext" href="http://planetmath.org/encyclopedia/FunctionalEquationOfTheRiemannZetaFunction.html" rel="nofollow"&gt;functional equation&lt;/a&gt; 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.&lt;br /&gt;
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 &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/CriticalStrip.html" rel="nofollow"&gt;critical strip&lt;/a&gt;, when s lies between zero and one. As s approaches the value s=1/2 of the &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/CriticalLine.html" rel="nofollow"&gt;critical line&lt;/a&gt;, 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 &lt;a class="wiki_link_ext" href="http://planetmath.org/encyclopedia/FunctionalEquationOfTheRiemannZetaFunction.html" rel="nofollow"&gt;functional equation&lt;/a&gt; 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.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The question now arises as to how we can continue the zeta function into the critical strip in a way which supports the connection with tuning and the claims above. There are various approaches to this, but perhaps the simplest is the use of the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Dirichlet_eta_function" rel="nofollow"&gt;Dirichlet eta function&lt;/a&gt; which was introduced to mathematics by &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Johann_Peter_Gustav_Lejeune_Dirichlet" rel="nofollow"&gt;Johann Peter Gustav Lejeune Dirichlet&lt;/a&gt;, who despite his name was a German and the brother-in-law of &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Felix_Mendelssohn_Bartholdy" rel="nofollow"&gt;Felix Mendelssohn&lt;/a&gt;.&lt;br /&gt;
As s&amp;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 &amp;gt;&amp;gt; 1 and x is an integer, the real part of zeta is approximately 1 + 2^(-s), and the imaginary part is approximatly zero; that is, zeta is 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 &amp;gt;&amp;gt; 1 the derivative is approximately -ln(2)/2^s, it is negative, meaning that this real value for zeta increases as we go back. The zeta function approaches 1 uniformly as s increases to infinity, so going back, 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 &amp;quot;Gram points&amp;quot;, after &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/J%C3%B8rgen_Pedersen_Gram" rel="nofollow"&gt;Jørgen Pedersen Gram&lt;/a&gt;. We thus have associated pure-octave edos, where x is an integer, to a value near to the pure octave, at the special sort of Gram point which corresponds to an edo.&lt;br /&gt;
&lt;br /&gt;
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 &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Riemann" rel="nofollow"&gt;Bernhard Riemann&lt;/a&gt; which he failed to publish because it was so nasty becomes a bit simpler when used at a Gram point. It is named the &lt;a class="wiki_link" href="/Riemann-Siegel%20formula"&gt;Riemann-Siegel formula&lt;/a&gt; since &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Carl_Ludwig_Siegel" rel="nofollow"&gt;Carl Ludwig Siegel&lt;/a&gt; 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.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="The Z function"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;The Z function&lt;/h1&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Computing zeta"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Computing zeta&lt;/h1&gt;
There are various approaches to the question of computing the zeta function, but perhaps the simplest is the use of the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Dirichlet_eta_function" rel="nofollow"&gt;Dirichlet eta function&lt;/a&gt; which was introduced to mathematics by &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Johann_Peter_Gustav_Lejeune_Dirichlet" rel="nofollow"&gt;Johann Peter Gustav Lejeune Dirichlet&lt;/a&gt;, who despite his name was a German and the brother-in-law of &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Felix_Mendelssohn_Bartholdy" rel="nofollow"&gt;Felix Mendelssohn&lt;/a&gt;.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The zeta function has a &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/SimplePole.html" rel="nofollow"&gt;simple pole&lt;/a&gt; at z=1 which forms a barrier against continuing it with its &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Euler_product" rel="nofollow"&gt;Euler product&lt;/a&gt; or &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Dirichlet_series" rel="nofollow"&gt;Dirichlet series&lt;/a&gt; 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:&lt;br /&gt;
The zeta function has a &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/SimplePole.html" rel="nofollow"&gt;simple pole&lt;/a&gt; at z=1 which forms a barrier against continuing it with its &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Euler_product" rel="nofollow"&gt;Euler product&lt;/a&gt; or &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Dirichlet_series" rel="nofollow"&gt;Dirichlet series&lt;/a&gt; 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:&lt;br /&gt;
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The Dirichlet series for the zeta function is absolutely convergent when s&amp;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 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 &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Euler_summation" rel="nofollow"&gt;Euler summation&lt;/a&gt;.&lt;br /&gt;
The Dirichlet series for the zeta function is absolutely convergent when s&amp;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 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 &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Euler_summation" rel="nofollow"&gt;Euler summation&lt;/a&gt;.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
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As s&amp;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 &amp;gt;&amp;gt; 1 and x is an integer, the real part of zeta is approximately 1 + 2^(-s), and the imaginary part is approximatly zero; that is, zeta is 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 &amp;gt;&amp;gt; 1 the derivative is approximately -ln(2)/2^s, it is negative, meaning that this real value for zeta increases as we go back. The zeta function approaches 1 uniformly as s increases to infinity, so going back, 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 &amp;quot;Gram points&amp;quot;, after &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/J%C3%B8rgen_Pedersen_Gram" rel="nofollow"&gt;Jørgen Pedersen Gram&lt;/a&gt;. We thus have associated pure-octave edos, where x is an integer, to a value near to the pure octave, at the special sort of Gram point which corresponds to an edo.&lt;br /&gt;
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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 &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Riemann" rel="nofollow"&gt;Bernhard Riemann&lt;/a&gt; which he failed to publish because it was so nasty becomes a bit simpler when used at a Gram point. It is named the &lt;a class="wiki_link" href="/Riemann-Siegel%20formula"&gt;Riemann-Siegel formula&lt;/a&gt; since &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Carl_Ludwig_Siegel" rel="nofollow"&gt;Carl Ludwig Siegel&lt;/a&gt; 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.&lt;br /&gt;
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&lt;a class="wiki_link_ext" href="http://front.math.ucdavis.edu/0309.5433" rel="nofollow"&gt;X-Ray of Riemann zeta-function&lt;/a&gt; by Juan Arias-de-Reyna&lt;/body&gt;&lt;/html&gt;</pre></div>
&lt;a class="wiki_link_ext" href="http://front.math.ucdavis.edu/0309.5433" rel="nofollow"&gt;X-Ray of Riemann zeta-function&lt;/a&gt; by Juan Arias-de-Reyna&lt;/body&gt;&lt;/html&gt;</pre></div>
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