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>2012-07-16 16:03:56 UTC</tt>.<br>

: This revision was by author [[User:clumma|clumma]] and made on <tt>2012-07-16 16:25:52 UTC</tt>.<br>

: The original revision id was <tt>~~353388612~~</tt>.<br>

: The original revision id was <tt>353391526</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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=The Black Magic Formulas=

When [[Gene Ward Smith|Gene Smith]] discovered these formulas in the 70s, he thought of them as "black magic" formulas not because of any aura of evil, but because they seemed mysteriously to give you something for next to nothing. They are based on Gram points and the Riemann-Siegel theta function θ(t). Recall that a Gram point is a ~~points~~ on the critical line where ζ(1/2 + ig) is real. This implies that exp(iθ(g)) is real, so that θ(g)/π is an integer. Theta has an [[http://en.wikipedia.org/wiki/Asymptotic_expansion|asymptotic expansion]]

When [[Gene Ward Smith|Gene Smith]] discovered these formulas in the 70s, he thought of them as "black magic" formulas not because of any aura of evil, but because they seemed mysteriously to give you something for next to nothing. They are based on Gram points and the Riemann-Siegel theta function θ(t). Recall that a Gram point is a point on the critical line where ζ(1/2 + ig) is real. This implies that exp(iθ(g)) is real, so that θ(g)/π is an integer. Theta has an [[http://en.wikipedia.org/wiki/Asymptotic_expansion|asymptotic expansion]]

[[math]]

\theta(t) \sim \frac{t}{2}\log \frac{t}{2\pi} - \frac{t}{2} - \frac{\pi}{8}+\frac{1}{48t}+ \frac{7}{5760t^3}+\cdots

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<br />

<h1 id="toc5"><a name="The Black Magic Formulas"></a>The Black Magic Formulas</h1>

When <a class="wiki_link" href="/Gene%20Ward%20Smith">Gene Smith</a> discovered these formulas in the 70s, he thought of them as &quot;black magic&quot; formulas not because of any aura of evil, but because they seemed mysteriously to give you something for next to nothing. They are based on Gram points and the Riemann-Siegel theta function θ(t). Recall that a Gram point is a ~~points~~ on the critical line where ζ(1/2 + ig) is real. This implies that exp(iθ(g)) is real, so that θ(g)/π is an integer. Theta has an <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Asymptotic_expansion" rel="nofollow">asymptotic expansion</a><br />

When <a class="wiki_link" href="/Gene%20Ward%20Smith">Gene Smith</a> discovered these formulas in the 70s, he thought of them as &quot;black magic&quot; formulas not because of any aura of evil, but because they seemed mysteriously to give you something for next to nothing. They are based on Gram points and the Riemann-Siegel theta function θ(t). Recall that a Gram point is a point on the critical line where ζ(1/2 + ig) is real. This implies that exp(iθ(g)) is real, so that θ(g)/π is an integer. Theta has an <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Asymptotic_expansion" rel="nofollow">asymptotic expansion</a><br />

<!-- ws:start:WikiTextMathRule:13:

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

@@ 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>2012-07-16 16:03:56 UTC</tt>.<br>
+: This revision was by author [[User:clumma|clumma]] and made on <tt>2012-07-16 16:25:52 UTC</tt>.<br>
-: The original revision id was <tt>353388612</tt>.<br>
+: The original revision id was <tt>353391526</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 139: / Line 139: @@
 =The Black Magic Formulas=
-When [[Gene Ward Smith|Gene Smith]] discovered these formulas in the 70s, he thought of them as "black magic" formulas not because of any aura of evil, but because they seemed mysteriously to give you something for next to nothing. They are based on Gram points and the Riemann-Siegel theta function θ(t). Recall that a Gram point is a points on the critical line where ζ(1/2 + ig) is real. This implies that exp(iθ(g)) is real, so that θ(g)/π is an integer. Theta has an [[http://en.wikipedia.org/wiki/Asymptotic_expansion|asymptotic expansion]]
+When [[Gene Ward Smith|Gene Smith]] discovered these formulas in the 70s, he thought of them as "black magic" formulas not because of any aura of evil, but because they seemed mysteriously to give you something for next to nothing. They are based on Gram points and the Riemann-Siegel theta function θ(t). Recall that a Gram point is a point on the critical line where ζ(1/2 + ig) is real. This implies that exp(iθ(g)) is real, so that θ(g)/π is an integer. Theta has an [[http://en.wikipedia.org/wiki/Asymptotic_expansion|asymptotic expansion]]
 [[math]]
 \theta(t) \sim \frac{t}{2}\log \frac{t}{2\pi} - \frac{t}{2} - \frac{\pi}{8}+\frac{1}{48t}+ \frac{7}{5760t^3}+\cdots
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 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:25:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="The Black Magic Formulas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:25 --&gt;The Black Magic Formulas&lt;/h1&gt;
-When &lt;a class="wiki_link" href="/Gene%20Ward%20Smith"&gt;Gene Smith&lt;/a&gt; discovered these formulas in the 70s, he thought of them as &amp;quot;black magic&amp;quot; formulas not because of any aura of evil, but because they seemed mysteriously to give you something for next to nothing. They are based on Gram points and the Riemann-Siegel theta function θ(t). Recall that a Gram point is a points on the critical line where ζ(1/2 + ig) is real. This implies that exp(iθ(g)) is real, so that θ(g)/π is an integer. Theta has an &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Asymptotic_expansion" rel="nofollow"&gt;asymptotic expansion&lt;/a&gt;&lt;br /&gt;
+When &lt;a class="wiki_link" href="/Gene%20Ward%20Smith"&gt;Gene Smith&lt;/a&gt; discovered these formulas in the 70s, he thought of them as &amp;quot;black magic&amp;quot; formulas not because of any aura of evil, but because they seemed mysteriously to give you something for next to nothing. They are based on Gram points and the Riemann-Siegel theta function θ(t). Recall that a Gram point is a point on the critical line where ζ(1/2 + ig) is real. This implies that exp(iθ(g)) is real, so that θ(g)/π is an integer. Theta has an &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Asymptotic_expansion" rel="nofollow"&gt;asymptotic expansion&lt;/a&gt;&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:13:
 [[math]]&amp;lt;br/&amp;gt;