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>2011-04-10 15:11:35 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-04-10 15:43:37 UTC</tt>.<br>

: The original revision id was <tt>~~218891290~~</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.

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 ~~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 >> 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.

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 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, 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=

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]].

=Computing zeta=

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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.<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 ~~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 &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 sort of Gram point which corresponds to an edo.<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 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 &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 sort of Gram point which corresponds to an edo.<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" href="/Riemann-Siegel%20formula">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 />

<h1 id="toc2"><a name="The Z function"></a>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 />

<h1 id="toc3"><a name="Computing zeta"></a>Computing zeta</h1>

@@ 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>2011-04-10 15:11:35 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-04-10 15:43:37 UTC</tt>.<br>
-: The original revision id was <tt>218891290</tt>.<br>
+: The original revision id was <tt>218897896</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 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.
+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 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=
+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 &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 [[http://en.wikipedia.org/wiki/Z_function|Z function]].
 =Computing zeta=
@@ Line 136: / Line 139: @@
 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 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;
+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 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;
+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 &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Riemann_hypothesis" rel="nofollow"&gt;Riemann hypothesis&lt;/a&gt; is equivalent to the claim that all zeros of zeta'(s + i t) occur when s &amp;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 &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Z_function" rel="nofollow"&gt;Z function&lt;/a&gt;.&lt;br /&gt;
+&lt;br /&gt;
+&lt;br /&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;