The Riemann zeta function and tuning: Difference between revisions
Wikispaces>genewardsmith **Imported revision 218981610 - Original comment: ** |
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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> | 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 22: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-04-10 22:45:41 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>218988466</tt>.<br> | ||
: The revision comment was: <tt></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> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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[[math]] | [[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 online function plotter | Since the arctangent function is holomorphic in the strip with imaginary part between -1 and 1, it follows from the above formula, or arguing from the previous one, that theta is holomorphic in the strip with imaginary part between -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 expoenential 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. | |||
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[[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"><html><head><title>The Riemann Zeta Function and Tuning</title></head><body><!-- ws:start:WikiTextTocRule: | <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"><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: | <!-- 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 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 /> | 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 <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 /> | ||
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so that we see that the absolute value of the zeta function serves to measure the error of an equal division.<br /> | so that we see that the absolute value of the zeta function serves to measure the error of an equal division.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- 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 /> | 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 /> | ||
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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 <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 /> | 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 /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- 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 /> | 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 /> | ||
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- \arctan\left(\frac{2t}{4n+1}\right)\right)</script><!-- ws:end:WikiTextMathRule:9 --><br /> | - \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 online function plotter | Since the arctangent function is holomorphic in the strip with imaginary part between -1 and 1, it follows from the above formula, or arguing from the previous one, that theta is holomorphic in the strip with imaginary part between -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 /> | |||
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[[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 /> | |||
<br /> | |||
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 expoenential 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 /> | |||
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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 /> | 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 /> | 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 /> | ||
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<!-- ws:start:WikiTextMathRule: | <!-- ws:start:WikiTextMathRule:11: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
\eta(z) = (1-2^{1-z})\zeta(z) = \sum_{n=1}^\infty (-1)^{n-1} n^{-z}&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]] | = \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} | --><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: | = \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 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 /> | 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 <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: | <!-- 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</body></html></pre></div> | <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</body></html></pre></div> | ||