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:

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-04-10 14:44:18 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-04-10 14:47:33 UTC</tt>.

: The original revision id was <tt>~~218885676~~</tt>.

: The original revision id was <tt>218886326</tt>.

: The revision comment was: <tt></tt>

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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 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(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

[[math]]

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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 <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Continuous_function" rel="nofollow">~~discontinuous function|~~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

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

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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>2011-04-10 14:44:18 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-04-10 14:47:33 UTC</tt>.<br>
-: The original revision id was <tt>218885676</tt>.<br>
+: The original revision id was <tt>218886326</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 28: / Line 28: @@
 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 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(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
 [[math]]
@@ Line 103: / Line 103: @@
 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 function|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(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;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:3: