The Riemann zeta function and tuning: Difference between revisions
Wikispaces>genewardsmith **Imported revision 244790239 - 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- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-03 13:30:46 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>250507782</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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Finally, taking the midpoints of the successively larger normalized gaps between the zeros of Z leads to a list of //zeta gap edos//. These are 2, 3, 5, 7, 12, 19, 31, 46, 53, 72, 270, 311, 954, 1178, 1308, 1395, 1578, 3395, 4190 ... Since the density of the zeros increases logarithmically, the normalization is to divide through by the log of the midpoint. These edos are listed in the OEIS as [[http://oeis.org/A117537|sequence A117537]]. The zeta gap edos seem to weight higher primes more heavily and have the advantage of being easy to compute from a table of zeros on the critical line. | Finally, taking the midpoints of the successively larger normalized gaps between the zeros of Z leads to a list of //zeta gap edos//. These are 2, 3, 5, 7, 12, 19, 31, 46, 53, 72, 270, 311, 954, 1178, 1308, 1395, 1578, 3395, 4190 ... Since the density of the zeros increases logarithmically, the normalization is to divide through by the log of the midpoint. These edos are listed in the OEIS as [[http://oeis.org/A117537|sequence A117537]]. The zeta gap edos seem to weight higher primes more heavily and have the advantage of being easy to compute from a table of zeros on the critical line. | ||
=Removing primes= | |||
The [[http://mathworld.wolfram.com/EulerProduct.html|Euler product]] for the Riemann zeta function is | |||
[[math]] | |||
\zeta(s) = \prod_{p prime} (1 - p^{-s})^{-1} | |||
[[math]] | |||
It converges for values of s with real part greater than or equal to one, except for s=1 where it diverges to infinity. We may remove a finite list of primes from consideration by multiplying zeta(s) by the corresponding factors (1-p^(-s)) for each prime p we wish to remove. After we have done this, the smallest prime remaining will dominate peak values for s with large real part, and as before we can | |||
track these peaks backwards and, by analytical continuation, into the critical strip. In particular if we remove the prime 2, | |||
(1-2^(-s))zeta(s) is now dominated by 3, and the large peak values occur near equal divisions of the "tritave", ie 3. | |||
Along the critical line, |1 - p^(-1/2-i t)| may be written | |||
[[math]] | |||
1 + \frac{1}{p} - \frac{2 \cos(\ln p t)}{\sqrt{p}} | |||
[[math]] | |||
Multiplying the Z-function by this factor of adjustment gives a Z-function with the prime p removed from consideration. Zeta peak and zeta integral tunings may then be found as before. | |||
=Computing zeta= | =Computing zeta= | ||
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[[http://terrytao.wordpress.com/2009/07/12/selbergs-limit-theorem-for-the-riemann-zeta-function-on-the-critical-line/|Selberg's limit theorem]] by Terence Tao [[http://www.webcitation.org/5xrvgjW6T|Permalink]]</pre></div> | [[http://terrytao.wordpress.com/2009/07/12/selbergs-limit-theorem-for-the-riemann-zeta-function-on-the-critical-line/|Selberg's limit theorem]] by Terence Tao [[http://www.webcitation.org/5xrvgjW6T|Permalink]]</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:28:&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:28 --><!-- ws:start:WikiTextTocRule:29: --><a href="#Preliminaries">Preliminaries</a><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --> | <a href="#Into the critical strip">Into the critical strip</a><!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --> | <a href="#The Z function">The Z function</a><!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --> | <a href="#Zeta EDO lists">Zeta EDO lists</a><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --> | <a href="#Removing primes">Removing primes</a><!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --> | <a href="#Computing zeta">Computing zeta</a><!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --> | <a href="#Links">Links</a><!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc0"><a name="Preliminaries"></a><!-- ws:end:WikiTextHeadingRule:14 -->Preliminaries</h1> | ||
Suppose x is a continuous variable equal to the reciprocal of the step size of an equal division of the octave in fractions of an octave. For example, if the step size was 15 cents, then x = 1200/15 = 80, and we would be considering 80edo in pure octave tuning. If ||x|| denotes x minus x rounded to the nearest integer, or in other words the function x - floor(x+1/2), then the <a class="wiki_link" href="/Tenney-Euclidean%20metrics">Tenney-Euclidean error</a> for the <a class="wiki_link" href="/p-limit">p-limit</a> <a class="wiki_link" href="/val">val</a> obtained by rounding log2(q) x to the nearest integer for each prime q up to p will be<br /> | Suppose x is a continuous variable equal to the reciprocal of the step size of an equal division of the octave in fractions of an octave. For example, if the step size was 15 cents, then x = 1200/15 = 80, and we would be considering 80edo in pure octave tuning. If ||x|| denotes x minus x rounded to the nearest integer, or in other words the function x - floor(x+1/2), then the <a class="wiki_link" href="/Tenney-Euclidean%20metrics">Tenney-Euclidean error</a> for the <a class="wiki_link" href="/p-limit">p-limit</a> <a class="wiki_link" href="/val">val</a> obtained by rounding log2(q) x to the nearest integer for each prime q up to p will be<br /> | ||
<br /> | <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 /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc1"><a name="Into the critical strip"></a><!-- ws:end:WikiTextHeadingRule:16 -->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 /> | ||
<br /> | <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 /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:18:&lt;h1&gt; --><h1 id="toc2"><a name="The Z function"></a><!-- ws:end:WikiTextHeadingRule:18 -->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 /> | ||
<br /> | <br /> | ||
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If you have access to <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Mathematica" rel="nofollow">Mathematica</a>, which has Z, zeta and theta as a part of its suite of initially defined functions, you can do even better. Below is a Mathematicia-generated plot of Z(2 pi x /ln(2)) in the region around 12edo:<br /> | If you have access to <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Mathematica" rel="nofollow">Mathematica</a>, which has Z, zeta and theta as a part of its suite of initially defined functions, you can do even better. Below is a Mathematicia-generated plot of Z(2 pi x /ln(2)) in the region around 12edo:<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:37:&lt;img src=&quot;/file/view/plot12.png/219376858/plot12.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/plot12.png/219376858/plot12.png" alt="plot12.png" title="plot12.png" /><!-- ws:end:WikiTextLocalImageRule:37 --><br /> | ||
<br /> | <br /> | ||
The peak around 12 is both higher and wider than the local maximums above 11 and 13, indicating its superiority as an edo. Note also that the peak occurs at a point slightly larger than 12; this indicates the octave is slightly compressed in the zeta tuning for 12. The size of a step in octaves is 1/x, and hence the size of the octave in the zeta peak value tuning for Nedo is N/x; if x is slightly larger than N as here with N=12, the size of the zeta tuned octave will be slightly less than a pure octave. Similarly, when the peak occurs with x less than N, we have stretched octaves.<br /> | The peak around 12 is both higher and wider than the local maximums above 11 and 13, indicating its superiority as an edo. Note also that the peak occurs at a point slightly larger than 12; this indicates the octave is slightly compressed in the zeta tuning for 12. The size of a step in octaves is 1/x, and hence the size of the octave in the zeta peak value tuning for Nedo is N/x; if x is slightly larger than N as here with N=12, the size of the zeta tuned octave will be slightly less than a pure octave. Similarly, when the peak occurs with x less than N, we have stretched octaves.<br /> | ||
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For larger edos, the width of the peak narrows, but for strong edos the height more than compensates, measured in terms of the area under the peak (the absolute value of the integral of Z between two zeros.) Note how 270 completely dominates its neighbors:<br /> | For larger edos, the width of the peak narrows, but for strong edos the height more than compensates, measured in terms of the area under the peak (the absolute value of the integral of Z between two zeros.) Note how 270 completely dominates its neighbors:<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:38:&lt;img src=&quot;/file/view/plot270.png/219383970/plot270.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/plot270.png/219383970/plot270.png" alt="plot270.png" title="plot270.png" /><!-- ws:end:WikiTextLocalImageRule:38 --><br /> | ||
<br /> | <br /> | ||
Note that for one of its neighbors, 271, it isn't entirely clear which peak value corresponds to the line of real values from +infinity. This can be determined by looking at the absolute value of zeta along other s values, such as s=1 or s=3/4, and in this case the local minimum at 271.069 is the value in question. However, other peak values are not without their interest; the local maximum at 270.941, for instance, is associated to a different mapping for 3.<br /> | Note that for one of its neighbors, 271, it isn't entirely clear which peak value corresponds to the line of real values from +infinity. This can be determined by looking at the absolute value of zeta along other s values, such as s=1 or s=3/4, and in this case the local minimum at 271.069 is the value in question. However, other peak values are not without their interest; the local maximum at 270.941, for instance, is associated to a different mapping for 3.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:20:&lt;h1&gt; --><h1 id="toc3"><a name="Zeta EDO lists"></a><!-- ws:end:WikiTextHeadingRule:20 -->Zeta EDO lists</h1> | ||
If we examine the increasingly larger peak values of |Z(x)|, we find occur with values of x such that Z'(x) = 0 near to integers, so that there is a sequence of <a class="wiki_link" href="/edo">edo</a>s 1, 2, 3, 4, 5, 7, 10, 12, 19, 22, 27, 31, 41, 53, 72, 99, 118, 130, 152, 171, 217, 224, 270, 342, 422, 441, 494, 742, 764, 935, 954, 1012, 1106, 1178, 1236, 1395, 1448, 1578, 2460, 2684, 3395, 5585, 6079, 7033, 8269, 8539, 11664 ... of <em>zeta peak edos</em>. This is listed in the On-Line Encyclopedia of Integer Sequences as <a class="wiki_link_ext" href="http://oeis.org/A117536" rel="nofollow">sequence A117536</a>. <br /> | If we examine the increasingly larger peak values of |Z(x)|, we find occur with values of x such that Z'(x) = 0 near to integers, so that there is a sequence of <a class="wiki_link" href="/edo">edo</a>s 1, 2, 3, 4, 5, 7, 10, 12, 19, 22, 27, 31, 41, 53, 72, 99, 118, 130, 152, 171, 217, 224, 270, 342, 422, 441, 494, 742, 764, 935, 954, 1012, 1106, 1178, 1236, 1395, 1448, 1578, 2460, 2684, 3395, 5585, 6079, 7033, 8269, 8539, 11664 ... of <em>zeta peak edos</em>. This is listed in the On-Line Encyclopedia of Integer Sequences as <a class="wiki_link_ext" href="http://oeis.org/A117536" rel="nofollow">sequence A117536</a>. <br /> | ||
<br /> | <br /> | ||
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Finally, taking the midpoints of the successively larger normalized gaps between the zeros of Z leads to a list of <em>zeta gap edos</em>. These are 2, 3, 5, 7, 12, 19, 31, 46, 53, 72, 270, 311, 954, 1178, 1308, 1395, 1578, 3395, 4190 ... Since the density of the zeros increases logarithmically, the normalization is to divide through by the log of the midpoint. These edos are listed in the OEIS as <a class="wiki_link_ext" href="http://oeis.org/A117537" rel="nofollow">sequence A117537</a>. The zeta gap edos seem to weight higher primes more heavily and have the advantage of being easy to compute from a table of zeros on the critical line.<br /> | Finally, taking the midpoints of the successively larger normalized gaps between the zeros of Z leads to a list of <em>zeta gap edos</em>. These are 2, 3, 5, 7, 12, 19, 31, 46, 53, 72, 270, 311, 954, 1178, 1308, 1395, 1578, 3395, 4190 ... Since the density of the zeros increases logarithmically, the normalization is to divide through by the log of the midpoint. These edos are listed in the OEIS as <a class="wiki_link_ext" href="http://oeis.org/A117537" rel="nofollow">sequence A117537</a>. The zeta gap edos seem to weight higher primes more heavily and have the advantage of being easy to compute from a table of zeros on the critical line.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:22:&lt;h1&gt; --><h1 id="toc4"><a name="Removing primes"></a><!-- ws:end:WikiTextHeadingRule:22 -->Removing primes</h1> | ||
The <a class="wiki_link_ext" href="http://mathworld.wolfram.com/EulerProduct.html" rel="nofollow">Euler product</a> for the Riemann zeta function is<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextMathRule:11: | |||
[[math]]&lt;br/&gt; | |||
\zeta(s) = \prod_{p prime} (1 - p^{-s})^{-1}&lt;br/&gt;[[math]] | |||
--><script type="math/tex">\zeta(s) = \prod_{p prime} (1 - p^{-s})^{-1}</script><!-- ws:end:WikiTextMathRule:11 --><br /> | |||
<br /> | |||
It converges for values of s with real part greater than or equal to one, except for s=1 where it diverges to infinity. We may remove a finite list of primes from consideration by multiplying zeta(s) by the corresponding factors (1-p^(-s)) for each prime p we wish to remove. After we have done this, the smallest prime remaining will dominate peak values for s with large real part, and as before we can <br /> | |||
track these peaks backwards and, by analytical continuation, into the critical strip. In particular if we remove the prime 2, <br /> | |||
(1-2^(-s))zeta(s) is now dominated by 3, and the large peak values occur near equal divisions of the &quot;tritave&quot;, ie 3. <br /> | |||
<br /> | |||
Along the critical line, |1 - p^(-1/2-i t)| may be written<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextMathRule:12: | |||
[[math]]&lt;br/&gt; | |||
1 + \frac{1}{p} - \frac{2 \cos(\ln p t)}{\sqrt{p}}&lt;br/&gt;[[math]] | |||
--><script type="math/tex">1 + \frac{1}{p} - \frac{2 \cos(\ln p t)}{\sqrt{p}}</script><!-- ws:end:WikiTextMathRule:12 --><br /> | |||
<br /> | |||
Multiplying the Z-function by this factor of adjustment gives a Z-function with the prime p removed from consideration. Zeta peak and zeta integral tunings may then be found as before.<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:24:&lt;h1&gt; --><h1 id="toc5"><a name="Computing zeta"></a><!-- ws:end:WikiTextHeadingRule:24 -->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:13: | ||
[[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:13 --><br /> | ||
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The Dirichlet series for the zeta function is absolutely convergent when s&gt;1, justifying the rearrangement of terms leading to the alternating series for eta, which converges conditionally in the critical strip. The extra factor introduces zeros of the eta function at the points 1 + 2pi i x/ln(2) corresponding to pure octave divisions along the line s=1, but no other zeros, and in particular none in the critical strip and along the critical line. The convergence of the alternating series can be greatly accelerated by applying <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Euler_summation" rel="nofollow">Euler summation</a>.<br /> | The Dirichlet series for the zeta function is absolutely convergent when s&gt;1, justifying the rearrangement of terms leading to the alternating series for eta, which converges conditionally in the critical strip. The extra factor introduces zeros of the eta function at the points 1 + 2pi i x/ln(2) corresponding to pure octave divisions along the line s=1, but no other zeros, and in particular none in the critical strip and along the critical line. The convergence of the alternating series can be greatly accelerated by applying <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Euler_summation" rel="nofollow">Euler summation</a>.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:26:&lt;h1&gt; --><h1 id="toc6"><a name="Links"></a><!-- ws:end:WikiTextHeadingRule:26 -->Links</h1> | ||
<a class="wiki_link_ext" href="http://front.math.ucdavis.edu/0309.5433" rel="nofollow">X-Ray of Riemann zeta-function</a> by Juan Arias-de-Reyna<br /> | <a class="wiki_link_ext" href="http://front.math.ucdavis.edu/0309.5433" rel="nofollow">X-Ray of Riemann zeta-function</a> by Juan Arias-de-Reyna<br /> | ||
<a class="wiki_link_ext" href="http://terrytao.wordpress.com/2009/07/12/selbergs-limit-theorem-for-the-riemann-zeta-function-on-the-critical-line/" rel="nofollow">Selberg's limit theorem</a> by Terence Tao <a class="wiki_link_ext" href="http://www.webcitation.org/5xrvgjW6T" rel="nofollow">Permalink</a></body></html></pre></div> | <a class="wiki_link_ext" href="http://terrytao.wordpress.com/2009/07/12/selbergs-limit-theorem-for-the-riemann-zeta-function-on-the-critical-line/" rel="nofollow">Selberg's limit theorem</a> by Terence Tao <a class="wiki_link_ext" href="http://www.webcitation.org/5xrvgjW6T" rel="nofollow">Permalink</a></body></html></pre></div> | ||