The Riemann zeta function and tuning: Difference between revisions
Wikispaces>genewardsmith **Imported revision 296850066 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 314861008 - 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>2012- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-03-26 20:57:24 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>314861008</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]] | ||
E_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{1 - \cos(2 \pi x)}{n^s} | E_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{1 - \cos(2 \pi x \log_2 n)}{n^s} | ||
[[math]] | [[math]] | ||
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[[math]] | [[math]] | ||
F_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{\cos(2 \pi x)}{n^s} | F_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{\cos(2 \pi x \log_2 n)}{n^s} | ||
[[math]] | [[math]] | ||
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<!-- ws:start:WikiTextMathRule:3: | <!-- ws:start:WikiTextMathRule:3: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
E_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{1 - \cos(2 \pi x)}{n^s}&lt;br/&gt;[[math]] | E_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{1 - \cos(2 \pi x \log_2 n)}{n^s}&lt;br/&gt;[[math]] | ||
--><script type="math/tex">E_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{1 - \cos(2 \pi x)}{n^s}</script><!-- ws:end:WikiTextMathRule:3 --><br /> | --><script type="math/tex">E_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{1 - \cos(2 \pi x \log_2 n)}{n^s}</script><!-- ws:end:WikiTextMathRule:3 --><br /> | ||
<br /> | <br /> | ||
For any fixed s &gt; 1 this gives a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Analytic_function" rel="nofollow">real analytic function</a> defined for all x, and hence with all the smoothness properties we could desire. We can define essentially the same function by subtracting it from E_s(1/2)/2:<br /> | For any fixed s &gt; 1 this gives a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Analytic_function" rel="nofollow">real analytic function</a> defined for all x, and hence with all the smoothness properties we could desire. We can define essentially the same function by subtracting it from E_s(1/2)/2:<br /> | ||
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<!-- ws:start:WikiTextMathRule:4: | <!-- ws:start:WikiTextMathRule:4: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
F_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{\cos(2 \pi x)}{n^s}&lt;br/&gt;[[math]] | F_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{\cos(2 \pi x \log_2 n)}{n^s}&lt;br/&gt;[[math]] | ||
--><script type="math/tex">F_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{\cos(2 \pi x)}{n^s}</script><!-- ws:end:WikiTextMathRule:4 --><br /> | --><script type="math/tex">F_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{\cos(2 \pi x \log_2 n)}{n^s}</script><!-- ws:end:WikiTextMathRule:4 --><br /> | ||
<br /> | <br /> | ||
This now increases to a maximum value for low errors, rather than declining to a minimum. Of more interest is the fact that it is a known mathematical function, which can be expressed in terms of the real part of the logarithm of the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Riemann_zeta_function" rel="nofollow">Riemann zeta function</a>:<br /> | This now increases to a maximum value for low errors, rather than declining to a minimum. Of more interest is the fact that it is a known mathematical function, which can be expressed in terms of the real part of the logarithm of the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Riemann_zeta_function" rel="nofollow">Riemann zeta function</a>:<br /> | ||