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:<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-03 13:30:46 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-03 13:34:09 UTC</tt>.<br>

: The original revision id was <tt>~~250507782~~</tt>.<br>

: The original revision id was <tt>250508176</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>

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[[math]]

\zeta(s) = \~~prod_{p prime}~~ (1 - p^{-s})^{-1}

\zeta(s) = \prod_p (1 - p^{-s})^{-1}

[[math]]

It converges for values of s with real part greater than or equal to one, except for s=1 where it diverges to infinity. We may remove a finite list of primes from consideration by multiplying zeta(s) by the corresponding factors (1-p^(-s)) for each prime p we wish to remove. After we have done this, the smallest prime remaining will dominate peak values for s with large real part, and as before we can

where the product is over all primes p. The product converges for values of s with real part greater than or equal to one, except for s=1 where it diverges to infinity. We may remove a finite list of primes from consideration by multiplying zeta(s) by the corresponding factors (1-p^(-s)) for each prime p we wish to remove. After we have done this, the smallest prime remaining will dominate peak values for s with large real part, and as before we can track these peaks backwards and, by analytical continuation, into the critical strip. In particular if we remove the prime 2, (1-2^(-s))zeta(s) is now dominated by 3, and the large peak values occur near equal divisions of the "tritave", ie 3.

track these peaks backwards and, by analytical continuation, into the critical strip. In particular if we remove the prime 2,

(1-2^(-s))zeta(s) is now dominated by 3, and the large peak values occur near equal divisions of the "tritave", ie 3.

Along the critical line, |1 - p^(-1/2-i t)| may be written

[[math]]

1 + \frac{1}{p} - \frac{2 \cos(\ln p t)}{\sqrt{p}}

1 + \frac{1}{p} - \frac{2 \cos(\ln\ p t)}{\sqrt{p}}

[[math]]

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<!-- ws:start:WikiTextMathRule:11:

[[math]]&lt;br/&gt;

\zeta(s) = \~~prod_{p prime}~~ (1 - p^{-s})^{-1}&lt;br/&gt;[[math]]

\zeta(s) = \prod_p (1 - p^{-s})^{-1}&lt;br/&gt;[[math]]

--><script type="math/tex">\zeta(s) = \~~prod_{p prime}~~ (1 - p^{-s})^{-1}</script><br />

--><script type="math/tex">\zeta(s) = \prod_p (1 - p^{-s})^{-1}</script><br />

<br />

It converges for values of s with real part greater than or equal to one, except for s=1 where it diverges to infinity. We may remove a finite list of primes from consideration by multiplying zeta(s) by the corresponding factors (1-p^(-s)) for each prime p we wish to remove. After we have done this, the smallest prime remaining will dominate peak values for s with large real part, and as before we can ~~<br />~~

where the product is over all primes p. The product converges for values of s with real part greater than or equal to one, except for s=1 where it diverges to infinity. We may remove a finite list of primes from consideration by multiplying zeta(s) by the corresponding factors (1-p^(-s)) for each prime p we wish to remove. After we have done this, the smallest prime remaining will dominate peak values for s with large real part, and as before we can track these peaks backwards and, by analytical continuation, into the critical strip. In particular if we remove the prime 2, (1-2^(-s))zeta(s) is now dominated by 3, and the large peak values occur near equal divisions of the &quot;tritave&quot;, ie 3. <br />

track these peaks backwards and, by analytical continuation, into the critical strip. In particular if we remove the prime 2, ~~<br />~~

(1-2^(-s))zeta(s) is now dominated by 3, and the large peak values occur near equal divisions of the &quot;tritave&quot;, ie 3. <br />

<br />

Along the critical line, |1 - p^(-1/2-i t)| may be written<br />

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<!-- ws:start:WikiTextMathRule:12:

[[math]]&lt;br/&gt;

1 + \frac{1}{p} - \frac{2 \cos(\ln p t)}{\sqrt{p}}&lt;br/&gt;[[math]]

1 + \frac{1}{p} - \frac{2 \cos(\ln\ p t)}{\sqrt{p}}&lt;br/&gt;[[math]]

--><script type="math/tex">1 + \frac{1}{p} - \frac{2 \cos(\ln p t)}{\sqrt{p}}</script><br />

--><script type="math/tex">1 + \frac{1}{p} - \frac{2 \cos(\ln\ p t)}{\sqrt{p}}</script><br />

<br />

Multiplying the Z-function by this factor of adjustment gives a Z-function with the prime p removed from consideration. Zeta peak and zeta integral tunings may then be found as before.<br />

@@ 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>2011-09-03 13:30:46 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-03 13:34:09 UTC</tt>.<br>
-: The original revision id was <tt>250507782</tt>.<br>
+: The original revision id was <tt>250508176</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 119: / Line 119: @@
 [[math]]
-\zeta(s) = \prod_{p prime} (1 - p^{-s})^{-1}
+\zeta(s) = \prod_p (1 - p^{-s})^{-1}
 [[math]]
-It converges for values of s with real part greater than or equal to one, except for s=1 where it diverges to infinity. We may remove a finite list of primes from consideration by multiplying zeta(s) by the corresponding factors (1-p^(-s)) for each prime p we wish to remove. After we have done this, the smallest prime remaining will dominate peak values for s with large real part, and as before we can
+where the product is over all primes p. The product converges for values of s with real part greater than or equal to one, except for s=1 where it diverges to infinity. We may remove a finite list of primes from consideration by multiplying zeta(s) by the corresponding factors (1-p^(-s)) for each prime p we wish to remove. After we have done this, the smallest prime remaining will dominate peak values for s with large real part, and as before we can track these peaks backwards and, by analytical continuation, into the critical strip. In particular if we remove the prime 2, (1-2^(-s))zeta(s) is now dominated by 3, and the large peak values occur near equal divisions of the "tritave", ie 3.
-track these peaks backwards and, by analytical continuation, into the critical strip. In particular if we remove the prime 2,
-(1-2^(-s))zeta(s) is now dominated by 3, and the large peak values occur near equal divisions of the "tritave", ie 3.
 Along the critical line, |1 - p^(-1/2-i t)| may be written
 [[math]]
-+ \frac{1}{p} - \frac{2 \cos(\ln p t)}{\sqrt{p}}
++ \frac{1}{p} - \frac{2 \cos(\ln\ p t)}{\sqrt{p}}
 [[math]]
@@ Line 277: / Line 275: @@
 &lt;!-- ws:start:WikiTextMathRule:11:
 [[math]]&amp;lt;br/&amp;gt;
-\zeta(s) = \prod_{p prime} (1 - p^{-s})^{-1}&amp;lt;br/&amp;gt;[[math]]
+\zeta(s) = \prod_p (1 - p^{-s})^{-1}&amp;lt;br/&amp;gt;[[math]]
-  --&gt;&lt;script type="math/tex"&gt;\zeta(s) = \prod_{p prime} (1 - p^{-s})^{-1}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:11 --&gt;&lt;br /&gt;
+  --&gt;&lt;script type="math/tex"&gt;\zeta(s) = \prod_p (1 - p^{-s})^{-1}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:11 --&gt;&lt;br /&gt;
 &lt;br /&gt;
-It converges for values of s with real part greater than or equal to one, except for s=1 where it diverges to infinity. We may remove a finite list of primes from consideration by multiplying zeta(s) by the corresponding factors (1-p^(-s)) for each prime p we wish to remove. After we have done this, the smallest prime remaining will dominate peak values for s with large real part, and as before we can &lt;br /&gt;
+where the product is over all primes p. The product converges for values of s with real part greater than or equal to one, except for s=1 where it diverges to infinity. We may remove a finite list of primes from consideration by multiplying zeta(s) by the corresponding factors (1-p^(-s)) for each prime p we wish to remove. After we have done this, the smallest prime remaining will dominate peak values for s with large real part, and as before we can track these peaks backwards and, by analytical continuation, into the critical strip. In particular if we remove the prime 2, (1-2^(-s))zeta(s) is now dominated by 3, and the large peak values occur near equal divisions of the &amp;quot;tritave&amp;quot;, ie 3. &lt;br /&gt;
-track these peaks backwards and, by analytical continuation, into the critical strip. In particular if we remove the prime 2, &lt;br /&gt;
-(1-2^(-s))zeta(s) is now dominated by 3, and the large peak values occur near equal divisions of the &amp;quot;tritave&amp;quot;, ie 3. &lt;br /&gt;
 &lt;br /&gt;
 Along the critical line, |1 - p^(-1/2-i t)| may be written&lt;br /&gt;
@@ Line 288: / Line 284: @@
 &lt;!-- ws:start:WikiTextMathRule:12:
 [[math]]&amp;lt;br/&amp;gt;
-+ \frac{1}{p} - \frac{2 \cos(\ln p t)}{\sqrt{p}}&amp;lt;br/&amp;gt;[[math]]
++ \frac{1}{p} - \frac{2 \cos(\ln\ p t)}{\sqrt{p}}&amp;lt;br/&amp;gt;[[math]]
-  --&gt;&lt;script type="math/tex"&gt;1 + \frac{1}{p} - \frac{2 \cos(\ln p t)}{\sqrt{p}}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:12 --&gt;&lt;br /&gt;
+  --&gt;&lt;script type="math/tex"&gt;1 + \frac{1}{p} - \frac{2 \cos(\ln\ p t)}{\sqrt{p}}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:12 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 Multiplying the Z-function by this factor of adjustment gives a Z-function with the prime p removed from consideration. Zeta peak and zeta integral tunings may then be found as before.&lt;br /&gt;