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The Riemann zeta function and tuning: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
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=Zeta EDO lists=  
=Zeta EDO lists=  
If we examine the increasingly larger peak values of |Z(x)|, we find occur with values of x such that Z'(x) = 0 near to integers, so that there is a sequence of [[edo]]s [[1edo|1]], [[2edo|2]], [[3edo|3]], [[4edo|4]], [[5edo|5]], [[7edo|7]], [[10edo|10]], [[12edo|12]], [[19edo|19]], [[22edo|22]], [[27edo|27]], [[31edo|31]], [[41edo|41]], [[53edo|53]], [[72edo|72]], [[99edo|99]], [[118edo|118]], [[130edo|130]], [[152edo|152]], [[171edo|171]], 217, 224, 270, 342, 422, 441, 494, 742, 764, 935, 954, 1012, 1106, 1178, 1236, 1395, 1448, 1578, 2460, 2684, 3395, 5585, 6079, 7033, 8269, 8539, 11664 ... of //zeta peak edos//. This is listed in the On-Line Encyclopedia of Integer Sequences as [[http://oeis.org/A117536|sequence A117536]].
If we examine the increasingly larger peak values of |Z(x)|, we find occur with values of x such that Z'(x) = 0 near to integers, so that there is a sequence of [[edo]]s [[1edo|1]], [[2edo|2]], [[3edo|3]], [[4edo|4]], [[5edo|5]], [[7edo|7]], [[10edo|10]], [[12edo|12]], [[19edo|19]], [[22edo|22]], [[27edo|27]], [[31edo|31]], [[41edo|41]], [[53edo|53]], [[72edo|72]], [[99edo|99]], [[118edo|118]], [[130edo|130]], [[152edo|152]], [[171edo|171]], [[217edo|217]], [[224edo|224]], [[270edo|270]], [[342edo|342]], [[422edo|422]], [[441edo|441]], [[494edo|494]], [[742edo|742]], [[764edo|764]], [[935edo|935]], [[954edo|954]], [[1012edo|1012]], [[1106edo|1106]], [[1178edo|1178]], [[1236edo|1236]], [[1395edo|1395]], [[1448edo|1448]], [[1578edo|1578]], [[246oedo|2460]], [[2684edo|2684]], [[3395edo|3395]], [[5585edo|5585]], [[6079edo|6079]], [[7033edo|7033]], [[8269edo|8269]], [[8539edo|8539]], [[11664edo|11664]] ... of //zeta peak edos//. This is listed in the On-Line Encyclopedia of Integer Sequences as [[http://oeis.org/A117536|sequence A117536]].


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.
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 [[2edo|2]], [[5edo|5]], [[7edo|7]], [[12edo|12]], [[19edo|19]], [[31edo|31]], [[41edo|41]], [[53edo|53]], [[72edo|72]], [[130edo|130]], [[171edo|171]], [[224edo|224]], [[270edo|270]], [[764edo|764]], [[954edo|954]], [[1178edo|1178]], [[1395edo|1395]], [[1578edo|1578]], [[2684edo|2684]], [[3395edo|3395]], [[7033edo|7033]], [[8269edo|8269]], [[8539edo|8539]], [[14348edo|14348]], [[16808edo|16808]], [[36269edo|36269]], [[58973edo|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 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 [[2edo|2]], [[3edo|3]], [[5edo|5]], [[7edo|7]], [[12edo|12]], [[19edo|19]], [[31edo|31]], [[46edo|46]], [[53edo|53]], [[72edo|72]], [[270edo|270]], [[311edo|311]], [[954edo|954]], [[1178edo|1178]], [[1308edo|1308]], [[1395edo|1395]], [[1578edo|1578]], [[3395edo|3395]], [[4190edo|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.


=Removing primes=  
=Removing primes=  
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&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Zeta EDO lists"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;Zeta EDO lists&lt;/h1&gt;
&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Zeta EDO lists"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;Zeta EDO lists&lt;/h1&gt;
  If we examine the increasingly larger peak values of |Z(x)|, we find occur with values of x such that Z'(x) = 0 near to integers, so that there is a sequence of &lt;a class="wiki_link" href="/edo"&gt;edo&lt;/a&gt;s &lt;a class="wiki_link" href="/1edo"&gt;1&lt;/a&gt;, &lt;a class="wiki_link" href="/2edo"&gt;2&lt;/a&gt;, &lt;a class="wiki_link" href="/3edo"&gt;3&lt;/a&gt;, &lt;a class="wiki_link" href="/4edo"&gt;4&lt;/a&gt;, &lt;a class="wiki_link" href="/5edo"&gt;5&lt;/a&gt;, &lt;a class="wiki_link" href="/7edo"&gt;7&lt;/a&gt;, &lt;a class="wiki_link" href="/10edo"&gt;10&lt;/a&gt;, &lt;a class="wiki_link" href="/12edo"&gt;12&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19&lt;/a&gt;, &lt;a class="wiki_link" href="/22edo"&gt;22&lt;/a&gt;, &lt;a class="wiki_link" href="/27edo"&gt;27&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/41edo"&gt;41&lt;/a&gt;, &lt;a class="wiki_link" href="/53edo"&gt;53&lt;/a&gt;, &lt;a class="wiki_link" href="/72edo"&gt;72&lt;/a&gt;, &lt;a class="wiki_link" href="/99edo"&gt;99&lt;/a&gt;, &lt;a class="wiki_link" href="/118edo"&gt;118&lt;/a&gt;, &lt;a class="wiki_link" href="/130edo"&gt;130&lt;/a&gt;, &lt;a class="wiki_link" href="/152edo"&gt;152&lt;/a&gt;, &lt;a class="wiki_link" href="/171edo"&gt;171&lt;/a&gt;, 217, 224, 270, 342, 422, 441, 494, 742, 764, 935, 954, 1012, 1106, 1178, 1236, 1395, 1448, 1578, 2460, 2684, 3395, 5585, 6079, 7033, 8269, 8539, 11664 ... of &lt;em&gt;zeta peak edos&lt;/em&gt;. This is listed in the On-Line Encyclopedia of Integer Sequences as &lt;a class="wiki_link_ext" href="http://oeis.org/A117536" rel="nofollow"&gt;sequence A117536&lt;/a&gt;.&lt;br /&gt;
  If we examine the increasingly larger peak values of |Z(x)|, we find occur with values of x such that Z'(x) = 0 near to integers, so that there is a sequence of &lt;a class="wiki_link" href="/edo"&gt;edo&lt;/a&gt;s &lt;a class="wiki_link" href="/1edo"&gt;1&lt;/a&gt;, &lt;a class="wiki_link" href="/2edo"&gt;2&lt;/a&gt;, &lt;a class="wiki_link" href="/3edo"&gt;3&lt;/a&gt;, &lt;a class="wiki_link" href="/4edo"&gt;4&lt;/a&gt;, &lt;a class="wiki_link" href="/5edo"&gt;5&lt;/a&gt;, &lt;a class="wiki_link" href="/7edo"&gt;7&lt;/a&gt;, &lt;a class="wiki_link" href="/10edo"&gt;10&lt;/a&gt;, &lt;a class="wiki_link" href="/12edo"&gt;12&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19&lt;/a&gt;, &lt;a class="wiki_link" href="/22edo"&gt;22&lt;/a&gt;, &lt;a class="wiki_link" href="/27edo"&gt;27&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/41edo"&gt;41&lt;/a&gt;, &lt;a class="wiki_link" href="/53edo"&gt;53&lt;/a&gt;, &lt;a class="wiki_link" href="/72edo"&gt;72&lt;/a&gt;, &lt;a class="wiki_link" href="/99edo"&gt;99&lt;/a&gt;, &lt;a class="wiki_link" href="/118edo"&gt;118&lt;/a&gt;, &lt;a class="wiki_link" href="/130edo"&gt;130&lt;/a&gt;, &lt;a class="wiki_link" href="/152edo"&gt;152&lt;/a&gt;, &lt;a class="wiki_link" href="/171edo"&gt;171&lt;/a&gt;, &lt;a class="wiki_link" href="/217edo"&gt;217&lt;/a&gt;, &lt;a class="wiki_link" href="/224edo"&gt;224&lt;/a&gt;, &lt;a class="wiki_link" href="/270edo"&gt;270&lt;/a&gt;, &lt;a class="wiki_link" href="/342edo"&gt;342&lt;/a&gt;, &lt;a class="wiki_link" href="/422edo"&gt;422&lt;/a&gt;, &lt;a class="wiki_link" href="/441edo"&gt;441&lt;/a&gt;, &lt;a class="wiki_link" href="/494edo"&gt;494&lt;/a&gt;, &lt;a class="wiki_link" href="/742edo"&gt;742&lt;/a&gt;, &lt;a class="wiki_link" href="/764edo"&gt;764&lt;/a&gt;, &lt;a class="wiki_link" href="/935edo"&gt;935&lt;/a&gt;, &lt;a class="wiki_link" href="/954edo"&gt;954&lt;/a&gt;, &lt;a class="wiki_link" href="/1012edo"&gt;1012&lt;/a&gt;, &lt;a class="wiki_link" href="/1106edo"&gt;1106&lt;/a&gt;, &lt;a class="wiki_link" href="/1178edo"&gt;1178&lt;/a&gt;, &lt;a class="wiki_link" href="/1236edo"&gt;1236&lt;/a&gt;, &lt;a class="wiki_link" href="/1395edo"&gt;1395&lt;/a&gt;, &lt;a class="wiki_link" href="/1448edo"&gt;1448&lt;/a&gt;, &lt;a class="wiki_link" href="/1578edo"&gt;1578&lt;/a&gt;, &lt;a class="wiki_link" href="/246oedo"&gt;2460&lt;/a&gt;, &lt;a class="wiki_link" href="/2684edo"&gt;2684&lt;/a&gt;, &lt;a class="wiki_link" href="/3395edo"&gt;3395&lt;/a&gt;, &lt;a class="wiki_link" href="/5585edo"&gt;5585&lt;/a&gt;, &lt;a class="wiki_link" href="/6079edo"&gt;6079&lt;/a&gt;, &lt;a class="wiki_link" href="/7033edo"&gt;7033&lt;/a&gt;, &lt;a class="wiki_link" href="/8269edo"&gt;8269&lt;/a&gt;, &lt;a class="wiki_link" href="/8539edo"&gt;8539&lt;/a&gt;, &lt;a class="wiki_link" href="/11664edo"&gt;11664&lt;/a&gt; ... of &lt;em&gt;zeta peak edos&lt;/em&gt;. This is listed in the On-Line Encyclopedia of Integer Sequences as &lt;a class="wiki_link_ext" href="http://oeis.org/A117536" rel="nofollow"&gt;sequence A117536&lt;/a&gt;.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
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;
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 &lt;a class="wiki_link" href="/2edo"&gt;2&lt;/a&gt;, &lt;a class="wiki_link" href="/5edo"&gt;5&lt;/a&gt;, &lt;a class="wiki_link" href="/7edo"&gt;7&lt;/a&gt;, &lt;a class="wiki_link" href="/12edo"&gt;12&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/41edo"&gt;41&lt;/a&gt;, &lt;a class="wiki_link" href="/53edo"&gt;53&lt;/a&gt;, &lt;a class="wiki_link" href="/72edo"&gt;72&lt;/a&gt;, &lt;a class="wiki_link" href="/130edo"&gt;130&lt;/a&gt;, &lt;a class="wiki_link" href="/171edo"&gt;171&lt;/a&gt;, &lt;a class="wiki_link" href="/224edo"&gt;224&lt;/a&gt;, &lt;a class="wiki_link" href="/270edo"&gt;270&lt;/a&gt;, &lt;a class="wiki_link" href="/764edo"&gt;764&lt;/a&gt;, &lt;a class="wiki_link" href="/954edo"&gt;954&lt;/a&gt;, &lt;a class="wiki_link" href="/1178edo"&gt;1178&lt;/a&gt;, &lt;a class="wiki_link" href="/1395edo"&gt;1395&lt;/a&gt;, &lt;a class="wiki_link" href="/1578edo"&gt;1578&lt;/a&gt;, &lt;a class="wiki_link" href="/2684edo"&gt;2684&lt;/a&gt;, &lt;a class="wiki_link" href="/3395edo"&gt;3395&lt;/a&gt;, &lt;a class="wiki_link" href="/7033edo"&gt;7033&lt;/a&gt;, &lt;a class="wiki_link" href="/8269edo"&gt;8269&lt;/a&gt;, &lt;a class="wiki_link" href="/8539edo"&gt;8539&lt;/a&gt;, &lt;a class="wiki_link" href="/14348edo"&gt;14348&lt;/a&gt;, &lt;a class="wiki_link" href="/16808edo"&gt;16808&lt;/a&gt;, &lt;a class="wiki_link" href="/36269edo"&gt;36269&lt;/a&gt;, &lt;a class="wiki_link" href="/58973edo"&gt;58973&lt;/a&gt; ... 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;
&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;
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 &lt;a class="wiki_link" href="/2edo"&gt;2&lt;/a&gt;, &lt;a class="wiki_link" href="/3edo"&gt;3&lt;/a&gt;, &lt;a class="wiki_link" href="/5edo"&gt;5&lt;/a&gt;, &lt;a class="wiki_link" href="/7edo"&gt;7&lt;/a&gt;, &lt;a class="wiki_link" href="/12edo"&gt;12&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/46edo"&gt;46&lt;/a&gt;, &lt;a class="wiki_link" href="/53edo"&gt;53&lt;/a&gt;, &lt;a class="wiki_link" href="/72edo"&gt;72&lt;/a&gt;, &lt;a class="wiki_link" href="/270edo"&gt;270&lt;/a&gt;, &lt;a class="wiki_link" href="/311edo"&gt;311&lt;/a&gt;, &lt;a class="wiki_link" href="/954edo"&gt;954&lt;/a&gt;, &lt;a class="wiki_link" href="/1178edo"&gt;1178&lt;/a&gt;, &lt;a class="wiki_link" href="/1308edo"&gt;1308&lt;/a&gt;, &lt;a class="wiki_link" href="/1395edo"&gt;1395&lt;/a&gt;, &lt;a class="wiki_link" href="/1578edo"&gt;1578&lt;/a&gt;, &lt;a class="wiki_link" href="/3395edo"&gt;3395&lt;/a&gt;, &lt;a class="wiki_link" href="/4190edo"&gt;4190&lt;/a&gt; ... 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;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Removing primes"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;Removing primes&lt;/h1&gt;
&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Removing primes"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;Removing primes&lt;/h1&gt;
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