The Riemann zeta function and tuning: Difference between revisions

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>2012-07-15 22:21:04 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-07-15 22:22:29 UTC</tt>.<br>

: The original revision id was <tt>~~353276468~~</tt>.<br>

: The original revision id was <tt>353276654</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 32:

where the summation is taken formally over all positive integers, though only the primes and prime powers make a nonzero contribution.

Another consequence of the above definition which might be objected to is that it results in a function with a [[http://en.wikipedia.org/wiki/Continuous_function|discontinuous derivative]], whereas a smooth function be preferred. The function ||x||^2 is quadratically increasing near integer values of x, and is periodic with period 1. Another function with these same properties is 1 - cos(~~2 pi x~~), which is a smooth and in fact an [[http://en.wikipedia.org/wiki/Entire_function|entire]] function. Let us therefore now define for any s > 1

Another consequence of the above definition which might be objected to is that it results in a function with a [[http://en.wikipedia.org/wiki/Continuous_function|discontinuous derivative]], whereas a smooth function be preferred. The function ||x||^2 is quadratically increasing near integer values of x, and is periodic with period 1. Another function with these same properties is 1 - cos(2πx), which is a smooth and in fact an [[http://en.wikipedia.org/wiki/Entire_function|entire]] function. Let us therefore now define for any s > 1

[[math]]

Line 148:

[[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~~/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 [[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 + 2πix/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 [[http://en.wikipedia.org/wiki/Euler_summation|Euler summation]].

=Links=

Line 183:

where the summation is taken formally over all positive integers, though only the primes and prime powers make a nonzero contribution.<br />

<br />

Another consequence of the above definition which might be objected to is that it results in a function with a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Continuous_function" rel="nofollow">discontinuous derivative</a>, whereas a smooth function be preferred. The function ||x||^2 is quadratically increasing near integer values of x, and is periodic with period 1. Another function with these same properties is 1 - cos(~~2 pi x~~), which is a smooth and in fact an <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Entire_function" rel="nofollow">entire</a> function. Let us therefore now define for any s &gt; 1<br />

Another consequence of the above definition which might be objected to is that it results in a function with a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Continuous_function" rel="nofollow">discontinuous derivative</a>, whereas a smooth function be preferred. The function ||x||^2 is quadratically increasing near integer values of x, and is periodic with period 1. Another function with these same properties is 1 - cos(2πx), which is a smooth and in fact an <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Entire_function" rel="nofollow">entire</a> function. Let us therefore now define for any s &gt; 1<br />

<br />

<!-- ws:start:WikiTextMathRule:3:

Line 313:

= \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~~/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 + 2πix/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 />

<br />

<h1 id="toc6"><a name="Links"></a>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://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>

@@ 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>2012-07-15 22:21:04 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-07-15 22:22:29 UTC</tt>.<br>
-: The original revision id was <tt>353276468</tt>.<br>
+: The original revision id was <tt>353276654</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 32: / Line 32: @@
 where the summation is taken formally over all positive integers, though only the primes and prime powers make a nonzero contribution.
-Another consequence of the above definition which might be objected to is that it results in a function with a [[http://en.wikipedia.org/wiki/Continuous_function|discontinuous derivative]], whereas a smooth function be preferred. The function ||x||^2 is quadratically increasing near integer values of x, and is periodic with period 1. Another function with these same properties is 1 - cos(2 pi x), which is a smooth and in fact an [[http://en.wikipedia.org/wiki/Entire_function|entire]] function. Let us therefore now define for any s &gt; 1
+Another consequence of the above definition which might be objected to is that it results in a function with a [[http://en.wikipedia.org/wiki/Continuous_function|discontinuous derivative]], whereas a smooth function be preferred. The function ||x||^2 is quadratically increasing near integer values of x, and is periodic with period 1. Another function with these same properties is 1 - cos(2πx), which is a smooth and in fact an [[http://en.wikipedia.org/wiki/Entire_function|entire]] function. Let us therefore now define for any s &gt; 1
 [[math]]
@@ Line 148: / Line 148: @@
 [[math]]
-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 [[http://en.wikipedia.org/wiki/Euler_summation|Euler summation]].
+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 + 2πix/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 [[http://en.wikipedia.org/wiki/Euler_summation|Euler summation]].
 =Links=
@@ Line 183: / Line 183: @@
 where the summation is taken formally over all positive integers, though only the primes and prime powers make a nonzero contribution.&lt;br /&gt;
 &lt;br /&gt;
-Another consequence of the above definition which might be objected to is that it results in a function with a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Continuous_function" rel="nofollow"&gt;discontinuous derivative&lt;/a&gt;, whereas a smooth function be preferred. The function ||x||^2 is quadratically increasing near integer values of x, and is periodic with period 1. Another function with these same properties is 1 - cos(2 pi x), which is a smooth and in fact an &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Entire_function" rel="nofollow"&gt;entire&lt;/a&gt; function. Let us therefore now define for any s &amp;gt; 1&lt;br /&gt;
+Another consequence of the above definition which might be objected to is that it results in a function with a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Continuous_function" rel="nofollow"&gt;discontinuous derivative&lt;/a&gt;, whereas a smooth function be preferred. The function ||x||^2 is quadratically increasing near integer values of x, and is periodic with period 1. Another function with these same properties is 1 - cos(2πx), which is a smooth and in fact an &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Entire_function" rel="nofollow"&gt;entire&lt;/a&gt; function. Let us therefore now define for any s &amp;gt; 1&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:3:
@@ Line 313: / Line 313: @@
 = \frac{1}{1^z} - \frac{1}{2^z} + \frac{1}{3^z} - \frac{1}{4^z} + \cdots&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:13 --&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/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 &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Euler_summation" rel="nofollow"&gt;Euler summation&lt;/a&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 + 2πix/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 &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Euler_summation" rel="nofollow"&gt;Euler summation&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Links"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;Links&lt;/h1&gt;
   &lt;a class="wiki_link_ext" href="http://front.math.ucdavis.edu/0309.5433" rel="nofollow"&gt;X-Ray of Riemann zeta-function&lt;/a&gt; by Juan Arias-de-Reyna&lt;br /&gt;
 &lt;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"&gt;Selberg's limit theorem&lt;/a&gt; by Terence Tao &lt;a class="wiki_link_ext" href="http://www.webcitation.org/5xrvgjW6T" rel="nofollow"&gt;Permalink&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>