The Riemann zeta function and tuning: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-04-11 ~~23:42~~:19 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-04-12 15:11:35 UTC</tt>.

: The original revision id was <tt>~~219384010~~</tt>.

: The original revision id was <tt>219624288</tt>.

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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.

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.

As s>1 gets larger, the Dirichlet series for the zeta function is increasingly dominated by the 2 term, getting ever closer to simply 1 + 2^(-z), which approaches 1 as s = Re(z) becomes larger. When s >> 1 and x is an integer, the real part of zeta is approximately 1 + 2^(-s), and the imaginary part is approximately zero; that is, zeta is approximately real. Starting from +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.

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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 <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.

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.

As s&gt;1 gets larger, the Dirichlet series for the zeta function is increasingly dominated by the 2 term, getting ever closer to simply 1 + 2^(-z), which approaches 1 as s = Re(z) becomes larger. When s &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.

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.

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-04-11 23:42:19 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-04-12 15:11:35 UTC</tt>.<br>
-: The original revision id was <tt>219384010</tt>.<br>
+: The original revision id was <tt>219624288</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
@@ Line 57: / Line 57: @@
 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&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 "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.
+As s&gt;1 gets larger, the Dirichlet series for the zeta function is increasingly dominated by the 2 term, getting ever closer to simply 1 + 2^(-z), which approaches 1 as s = Re(z) becomes larger. When s &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 "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.
@@ Line 180: / Line 180: @@
 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;
-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 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 &amp;gt;&amp;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 &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 sorts of Gram points which corresponds to edos.&lt;br /&gt;
+As s&amp;gt;1 gets larger, the Dirichlet series for the zeta function is increasingly dominated by the 2 term, getting ever closer to simply 1 + 2^(-z), which approaches 1 as s = Re(z) becomes larger. When s &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 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 &amp;gt;&amp;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 &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 sorts of Gram points which corresponds to edos.&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_ext" href="http://en.wikipedia.org/wiki/Riemann%E2%80%93Siegel_formula" rel="nofollow"&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;