The Riemann zeta function and tuning: Difference between revisions
Wikispaces>genewardsmith **Imported revision 238826637 - Original comment: ** |
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<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- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-08-08 01:15:19 UTC</tt>.<br> | ||
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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 1, 2, 3, 4, 5, 7, 10, 12, 19, 22, 27, 31, 41, 53, 72, 99, 118, 130, 152, 171, 217, 224, 270, 342, 422, 441, 494 ... 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 1, 2, 3, 4, 5, 7, 10, 12, 19, 22, 27, 31, 41, 53, 72, 99, 118, 130, 152, 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]]. | ||
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 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. | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:18:&lt;h1&gt; --><h1 id="toc3"><a name="Zeta EDO lists"></a><!-- ws:end:WikiTextHeadingRule:18 -->Zeta EDO lists</h1> | <!-- ws:start:WikiTextHeadingRule:18:&lt;h1&gt; --><h1 id="toc3"><a name="Zeta EDO lists"></a><!-- ws:end:WikiTextHeadingRule:18 -->Zeta EDO lists</h1> | ||
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 <a class="wiki_link" href="/edo">edo</a>s 1, 2, 3, 4, 5, 7, 10, 12, 19, 22, 27, 31, 41, 53, 72, 99, 118, 130, 152, 171, 217, 224, 270, 342, 422, 441, 494 ... of <em>zeta peak edos</em>. This is listed in the On-Line Encyclopedia of Integer Sequences as <a class="wiki_link_ext" href="http://oeis.org/A117536" rel="nofollow">sequence A117536</a>. <br /> | 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 <a class="wiki_link" href="/edo">edo</a>s 1, 2, 3, 4, 5, 7, 10, 12, 19, 22, 27, 31, 41, 53, 72, 99, 118, 130, 152, 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 <em>zeta peak edos</em>. This is listed in the On-Line Encyclopedia of Integer Sequences as <a class="wiki_link_ext" href="http://oeis.org/A117536" rel="nofollow">sequence A117536</a>. <br /> | ||
<br /> | <br /> | ||
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 /> | 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 /> | ||