The Riemann zeta function and tuning: Difference between revisions

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<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>2011-04-07 13:30:34 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-04-07 13:46:00 UTC</tt>.<br>

: The original revision id was <tt>~~218179040~~</tt>.<br>

: The original revision id was <tt>218185546</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>

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[[math]]

\eta(z) = (1-2^{1-z})\zeta(s) = \sum_{n=1}^\infty (-1)^{n-1} n^{-z}

\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

[[math]]

</pre></div>

The Dirichlet series for the zeta function is absolutely convergent when s>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.</pre></div>

<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><a href="#Preliminaries">Preliminaries</a> | <a href="#Into the critical strip">Into the critical strip</a>

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<!-- ws:start:WikiTextMathRule:7:

[[math]]&lt;br/&gt;

\eta(z) = (1-2^{1-z})\zeta(s) = \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]]

--><script type="math/tex">\eta(z) = (1-2^{1-z})\zeta(s) = \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></body></html></pre></div>

= \frac{1}{1^z} - \frac{1}{2^z} + \frac{1}{3^z} - \frac{1}{4^z} + \cdots</script><br />

<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.</body></html></pre></div>

@@ 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>2011-04-07 13:30:34 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-04-07 13:46:00 UTC</tt>.<br>
-: The original revision id was <tt>218179040</tt>.<br>
+: The original revision id was <tt>218185546</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 62: / Line 62: @@
 [[math]]
-\eta(z) = (1-2^{1-z})\zeta(s) = \sum_{n=1}^\infty (-1)^{n-1} n^{-z}
+\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
 [[math]]
-</pre></div>
+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.</pre></div>
 <h4>Original HTML content:</h4>
 <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;The Riemann Zeta Function and Tuning&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:12:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;&lt;a href="#Preliminaries"&gt;Preliminaries&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;!-- ws:start:WikiTextTocRule:14: --&gt; | &lt;a href="#Into the critical strip"&gt;Into the critical strip&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt;
@@ Line 132: / Line 132: @@
 &lt;!-- ws:start:WikiTextMathRule:7:
 [[math]]&amp;lt;br/&amp;gt;
-\eta(z) = (1-2^{1-z})\zeta(s) = \sum_{n=1}^\infty (-1)^{n-1} n^{-z}&amp;lt;br /&amp;gt;
+\eta(z) = (1-2^{1-z})\zeta(z) = \sum_{n=1}^\infty (-1)^{n-1} n^{-z}&amp;lt;br /&amp;gt;
 = \frac{1}{1^z} - \frac{1}{2^z} + \frac{1}{3^z} - \frac{1}{4^z} + \cdots&amp;lt;br/&amp;gt;[[math]]
-  --&gt;&lt;script type="math/tex"&gt;\eta(z) = (1-2^{1-z})\zeta(s) = \sum_{n=1}^\infty (-1)^{n-1} n^{-z}
+  --&gt;&lt;script type="math/tex"&gt;\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&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:7 --&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
+= \frac{1}{1^z} - \frac{1}{2^z} + \frac{1}{3^z} - \frac{1}{4^z} + \cdots&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:7 --&gt;&lt;br /&gt;
+&lt;br /&gt;
+The Dirichlet series for the zeta function is absolutely convergent when s&amp;gt;1, justifying the rearrangement of terms leading to the alternating series for eta, which converges conditionally in the critical strip. The extra factor introduces zeros of the eta function at the points 1 + 2pi i x corresponding to pure octave divisions along the line s=1, but no other zeros, and in particular none in the critical strip and along the critical line.&lt;/body&gt;&lt;/html&gt;</pre></div>