The Riemann zeta function and tuning: Difference between revisions
Wikispaces>genewardsmith **Imported revision 218988466 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 219003482 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<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- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-04-11 00:01:04 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>219003482</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> | ||
| Line 92: | Line 92: | ||
[[math]] | [[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 | 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 exponential 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. | ||
Using the [[http://functions.wolfram.com/webMathematica/FunctionPlotting.jsp?name=RiemannSiegelZ|online plotter]] we can plot Z in the regions corresponding to scale divisions, using the conversion factor t = 2pi/ln(2) x, for x a number near or at an edo number. Hence, for instance, to plot 12 plot around 108.777, to plot 31 plot around 281.006, and so forth. | |||
| Line 109: | Line 111: | ||
[[math]] | [[math]] | ||
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. The convergence of the alternating series can be greatly accelerated by applying [[http://en.wikipedia.org/wiki/Euler_summation|Euler summation]]. | 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/ln(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 [[http://en.wikipedia.org/wiki/Euler_summation|Euler summation]]. | ||
=Links= | =Links= | ||
| Line 213: | Line 215: | ||
--><script type="math/tex">Z(t) = \exp(i \theta(t)) \zeta(1/2 + it)</script><!-- ws:end:WikiTextMathRule:10 --><br /> | --><script type="math/tex">Z(t) = \exp(i \theta(t)) \zeta(1/2 + it)</script><!-- ws:end:WikiTextMathRule:10 --><br /> | ||
<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 | 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 exponential 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 /> | ||
<br /> | |||
Using the <a class="wiki_link_ext" href="http://functions.wolfram.com/webMathematica/FunctionPlotting.jsp?name=RiemannSiegelZ" rel="nofollow">online plotter</a> we can plot Z in the regions corresponding to scale divisions, using the conversion factor t = 2pi/ln(2) x, for x a number near or at an edo number. Hence, for instance, to plot 12 plot around 108.777, to plot 31 plot around 281.006, and so forth.<br /> | |||
<br /> | <br /> | ||
<br /> | <br /> | ||
| Line 232: | Line 236: | ||
= \frac{1}{1^z} - \frac{1}{2^z} + \frac{1}{3^z} - \frac{1}{4^z} + \cdots</script><!-- ws:end:WikiTextMathRule:11 --><br /> | = \frac{1}{1^z} - \frac{1}{2^z} + \frac{1}{3^z} - \frac{1}{4^z} + \cdots</script><!-- ws:end:WikiTextMathRule:11 --><br /> | ||
<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. 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(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 /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:20:&lt;h1&gt; --><h1 id="toc4"><a name="Links"></a><!-- ws:end:WikiTextHeadingRule:20 -->Links</h1> | <!-- 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> | ||