The Riemann zeta function and tuning: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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Similarly, if we take the integral of |Z(x)| between successive zeros, and use this to define a sequence of increasing values for this integral, these again occur near integers and define an edo. This sequence, the //zeta integral edos//, goes 2, 5, 7, 12, 19, 31, 41, 53, 72, 130, 171, 224, 270, 764, 954, 1178, 1395, 1578, 2684, 3395, 7033, 8269, 8539, 14348, 16808, 36269, 58973 ... This is listed in the OEIS as [[http://oeis.org/A117538|sequence A117538]]. The zeta integral edos seem to be, on the whole, the best of the zeta function sequences, but the other two should not be discounted; the peak values seem to give more weight to the lower primes, and the zeta gap sequence discussed below to the higher primes.

Finally, taking the midpoints of the successively larger normalized gaps between the zeros of Z leads to a list of //zeta ~~zero~~ gap edos//. These are 2, 3, 5, 7, 12, 19, 31, 46, 53, 72, 270, 311, 954, 1178, 1308, 1395, 1578, 3395, 4190 ... Since the density of the zeros increases logarithmically, the normalization is to divide through by the log of the midpoint. These edos are listed in the OEIS as [[http://oeis.org/A117537|sequence A117537]]. The zeta gap edos seem to weight higher primes more heavily and have the advantage of being easy to compute from a table of zeros on the critical line.

Finally, taking the midpoints of the successively larger normalized gaps between the zeros of Z leads to a list of //zeta gap edos//. These are 2, 3, 5, 7, 12, 19, 31, 46, 53, 72, 270, 311, 954, 1178, 1308, 1395, 1578, 3395, 4190 ... Since the density of the zeros increases logarithmically, the normalization is to divide through by the log of the midpoint. These edos are listed in the OEIS as [[http://oeis.org/A117537|sequence A117537]]. The zeta gap edos seem to weight higher primes more heavily and have the advantage of being easy to compute from a table of zeros on the critical line.

=Computing zeta=

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Similarly, if we take the integral of |Z(x)| between successive zeros, and use this to define a sequence of increasing values for this integral, these again occur near integers and define an edo. This sequence, the <em>zeta integral edos</em>, goes 2, 5, 7, 12, 19, 31, 41, 53, 72, 130, 171, 224, 270, 764, 954, 1178, 1395, 1578, 2684, 3395, 7033, 8269, 8539, 14348, 16808, 36269, 58973 ... This is listed in the OEIS as <a class="wiki_link_ext" href="http://oeis.org/A117538" rel="nofollow">sequence A117538</a>. The zeta integral edos seem to be, on the whole, the best of the zeta function sequences, but the other two should not be discounted; the peak values seem to give more weight to the lower primes, and the zeta gap sequence discussed below to the higher primes.<br />

<br />

Finally, taking the midpoints of the successively larger normalized gaps between the zeros of Z leads to a list of <em>zeta ~~zero~~ gap edos</em>. These are 2, 3, 5, 7, 12, 19, 31, 46, 53, 72, 270, 311, 954, 1178, 1308, 1395, 1578, 3395, 4190 ... Since the density of the zeros increases logarithmically, the normalization is to divide through by the log of the midpoint. These edos are listed in the OEIS as <a class="wiki_link_ext" href="http://oeis.org/A117537" rel="nofollow">sequence A117537</a>. The zeta gap edos seem to weight higher primes more heavily and have the advantage of being easy to compute from a table of zeros on the critical line.<br />

Finally, taking the midpoints of the successively larger normalized gaps between the zeros of Z leads to a list of <em>zeta gap edos</em>. These are 2, 3, 5, 7, 12, 19, 31, 46, 53, 72, 270, 311, 954, 1178, 1308, 1395, 1578, 3395, 4190 ... Since the density of the zeros increases logarithmically, the normalization is to divide through by the log of the midpoint. These edos are listed in the OEIS as <a class="wiki_link_ext" href="http://oeis.org/A117537" rel="nofollow">sequence A117537</a>. The zeta gap edos seem to weight higher primes more heavily and have the advantage of being easy to compute from a table of zeros on the critical line.<br />

<br />

<h1 id="toc4"><a name="Computing zeta"></a>Computing zeta</h1>

@@ 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>2011-06-26 15:00:05 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-06-26 15:03:15 UTC</tt>.<br>
-: The original revision id was <tt>238824585</tt>.<br>
+: The original revision id was <tt>238824825</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 114: / Line 114: @@
 Similarly, if we take the integral of |Z(x)| between successive zeros, and use this to define a sequence of increasing values for this integral, these again occur near integers and define an edo. This sequence, the //zeta integral edos//, goes 2, 5, 7, 12, 19, 31, 41, 53, 72, 130, 171, 224, 270, 764, 954, 1178, 1395, 1578, 2684, 3395, 7033, 8269, 8539, 14348, 16808, 36269, 58973 ... This is listed in the OEIS as [[http://oeis.org/A117538|sequence A117538]]. The zeta integral edos seem to be, on the whole, the best of the zeta function sequences, but the other two should not be discounted; the peak values seem to give more weight to the lower primes, and the zeta gap sequence discussed below to the higher primes.
-Finally, taking the midpoints of the successively larger normalized gaps between the zeros of Z leads to a list of //zeta zero gap edos//. These are 2, 3, 5, 7, 12, 19, 31, 46, 53, 72, 270, 311, 954, 1178, 1308, 1395, 1578, 3395, 4190 ... Since the density of the zeros increases logarithmically, the normalization is to divide through by the log of the midpoint. These edos are listed in the OEIS as [[http://oeis.org/A117537|sequence A117537]]. The zeta gap edos seem to weight higher primes more heavily and have the advantage of being easy to compute from a table of zeros on the critical line.
+Finally, taking the midpoints of the successively larger normalized gaps between the zeros of Z leads to a list of //zeta gap edos//. These are 2, 3, 5, 7, 12, 19, 31, 46, 53, 72, 270, 311, 954, 1178, 1308, 1395, 1578, 3395, 4190 ... Since the density of the zeros increases logarithmically, the normalization is to divide through by the log of the midpoint. These edos are listed in the OEIS as [[http://oeis.org/A117537|sequence A117537]]. The zeta gap edos seem to weight higher primes more heavily and have the advantage of being easy to compute from a table of zeros on the critical line.
 =Computing zeta=
@@ Line 253: / Line 253: @@
 Similarly, if we take the integral of |Z(x)| between successive zeros, and use this to define a sequence of increasing values for this integral, these again occur near integers and define an edo. This sequence, the &lt;em&gt;zeta integral edos&lt;/em&gt;, goes 2, 5, 7, 12, 19, 31, 41, 53, 72, 130, 171, 224, 270, 764, 954, 1178, 1395, 1578, 2684, 3395, 7033, 8269, 8539, 14348, 16808, 36269, 58973 ... This is listed in the OEIS as &lt;a class="wiki_link_ext" href="http://oeis.org/A117538" rel="nofollow"&gt;sequence A117538&lt;/a&gt;. The zeta integral edos seem to be, on the whole, the best of the zeta function sequences, but the other two should not be discounted; the peak values seem to give more weight to the lower primes, and the zeta gap sequence discussed below to the higher primes.&lt;br /&gt;
 &lt;br /&gt;
-Finally, taking the midpoints of the successively larger normalized gaps between the zeros of Z leads to a list of &lt;em&gt;zeta zero gap edos&lt;/em&gt;. These are 2, 3, 5, 7, 12, 19, 31, 46, 53, 72, 270, 311, 954, 1178, 1308, 1395, 1578, 3395, 4190 ... Since the density of the zeros increases logarithmically, the normalization is to divide through by the log of the midpoint. These edos are listed in the OEIS as &lt;a class="wiki_link_ext" href="http://oeis.org/A117537" rel="nofollow"&gt;sequence A117537&lt;/a&gt;. The zeta gap edos seem to weight higher primes more heavily and have the advantage of being easy to compute from a table of zeros on the critical line.&lt;br /&gt;
+Finally, taking the midpoints of the successively larger normalized gaps between the zeros of Z leads to a list of &lt;em&gt;zeta gap edos&lt;/em&gt;. These are 2, 3, 5, 7, 12, 19, 31, 46, 53, 72, 270, 311, 954, 1178, 1308, 1395, 1578, 3395, 4190 ... Since the density of the zeros increases logarithmically, the normalization is to divide through by the log of the midpoint. These edos are listed in the OEIS as &lt;a class="wiki_link_ext" href="http://oeis.org/A117537" rel="nofollow"&gt;sequence A117537&lt;/a&gt;. The zeta gap edos seem to weight higher primes more heavily and have the advantage of being easy to compute from a table of zeros on the critical line.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Computing zeta"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;Computing zeta&lt;/h1&gt;