The Riemann zeta function and tuning: Difference between revisions

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As s&gt;1 gets larger, the Dirichlet series for the zeta function is increasingly dominated by the 2 term, getting ever closer to simply 1 + 2^(-z), which approaches 1 as s = Re(z) becomes larger. When s &gt;&gt; 1 and x is an integer, the real part of zeta is approximately 1 + 2^(-s), and the imaginary part is approximately zero; that is, zeta is approximately real. Starting from +infinity with x an integer, we can trace a line back towards the critical strip on which zeta is real. Since when s &gt;&gt; 1 the derivative is approximately -ln(2)/2^s, it is negative on this line of real values for zeta, meaning that the real value for zeta increases as s decreases. The zeta function approaches 1 uniformly as s increases to infinity, so as s decreases, the real-valued zeta function along this line of real values continues to increase though all real values from 1 to infinity monotonically. When it crosses the critical line where s=1/2, it produces a real value of zeta on the critical line. Points on the critical line where zeta(1/2 + i g) are real are called "Gram points", after [[http://en.wikipedia.org/wiki/J%C3%B8rgen_Pedersen_Gram|Jørgen Pedersen Gram]]. We thus have associated pure-octave edos, where x is an integer, to a value near to the pure octave, at the special sorts of Gram points which corresponds to edos.

Because the value of zeta increased continuously as it made its way from +infinity to the critical line, we might expect the values of zeta at these special Gram points to be relatively large. This would be especially true if -zeta'(z) is getting a boost from other small primes as it travels toward the Gram point. A complex formula due to [[http://en.wikipedia.org/wiki/Riemann|Bernhard Riemann]] which he failed to publish because it was so nasty becomes a bit simpler when used at a Gram point. It is named the [[http://en.wikipedia.org/wiki/Riemann%E2%80%93Siegel_formula|Riemann-Siegel formula]] since [[http://en.wikipedia.org/wiki/Carl_Ludwig_Siegel|Carl Ludwig Siegel]] went looking for it and was able to reconstruct it after rooting industriously around in Riemann's unpublished papers. From this formula, it is apparent that when x corresponds to a good edo, the value of zeta(1/2 + i g) at the corresponding Gram point should be especially large.

The absolute value zeta(1/2 + i g) at a Gram point corresponding to an edo is near to a local maximum, but not actually at one. At the local maximum, of course, the partial derivative of zeta(1/2 + i t) with respect to t will be zero; however this does not mean its derivative there will be zero. In fact, the [[http://en.wikipedia.org/wiki/Riemann_hypothesis|Riemann hypothesis]] is equivalent to the claim that all zeros of zeta'(s + i t) occur when s &gt; 1/2, which is where all known zeros lie. These do not have values of t corresponding to good edos. For this and other reasons, it is helpful to have a function which is real for values on the critical line but whose absolute value is the same as that of zeta. This is provided by the [[http://en.wikipedia.org/wiki/Z_function|Z function]].

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.

As s&amp;gt;1 gets larger, the Dirichlet series for the zeta function is increasingly dominated by the 2 term, getting ever closer to simply 1 + 2^(-z), which approaches 1 as s = Re(z) becomes larger. When s &amp;gt;&amp;gt; 1 and x is an integer, the real part of zeta is approximately 1 + 2^(-s), and the imaginary part is approximately zero; that is, zeta is approximately real. Starting from +infinity with x an integer, we can trace a line back towards the critical strip on which zeta is real. Since when s &amp;gt;&amp;gt; 1 the derivative is approximately -ln(2)/2^s, it is negative on this line of real values for zeta, meaning that the real value for zeta increases as s decreases. The zeta function approaches 1 uniformly as s increases to infinity, so as s decreases, the real-valued zeta function along this line of real values continues to increase though all real values from 1 to infinity monotonically. When it crosses the critical line where s=1/2, it produces a real value of zeta on the critical line. Points on the critical line where zeta(1/2 + i g) are real are called &amp;quot;Gram points&amp;quot;, after &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/J%C3%B8rgen_Pedersen_Gram" rel="nofollow"&gt;Jørgen Pedersen Gram&lt;/a&gt;. We thus have associated pure-octave edos, where x is an integer, to a value near to the pure octave, at the special sorts of Gram points which corresponds to edos.&lt;br /&gt;

Because the value of zeta increased continuously as it made its way from +infinity to the critical line, we might expect the values of zeta at these special Gram points to be relatively large. This would be especially true if -zeta'(z) is getting a boost from other small primes as it travels toward the Gram point. A complex formula due to &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Riemann" rel="nofollow"&gt;Bernhard Riemann&lt;/a&gt; which he failed to publish because it was so nasty becomes a bit simpler when used at a Gram point. It is named the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Riemann%E2%80%93Siegel_formula" rel="nofollow"&gt;Riemann-Siegel formula&lt;/a&gt; since &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Carl_Ludwig_Siegel" rel="nofollow"&gt;Carl Ludwig Siegel&lt;/a&gt; went looking for it and was able to reconstruct it after rooting industriously around in Riemann's unpublished papers. From this formula, it is apparent that when x corresponds to a good edo, the value of zeta(1/2 + i g) at the corresponding Gram point should be especially large.&lt;br /&gt;

  The absolute value zeta(1/2 + i g) at a Gram point corresponding to an edo is near to a local maximum, but not actually at one. At the local maximum, of course, the partial derivative of zeta(1/2 + i t) with respect to t will be zero; however this does not mean its derivative there will be zero. In fact, the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Riemann_hypothesis" rel="nofollow"&gt;Riemann hypothesis&lt;/a&gt; is equivalent to the claim that all zeros of zeta'(s + i t) occur when s &amp;gt; 1/2, which is where all known zeros lie. These do not have values of t corresponding to good edos. For this and other reasons, it is helpful to have a function which is real for values on the critical line but whose absolute value is the same as that of zeta. This is provided by the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Z_function" rel="nofollow"&gt;Z function&lt;/a&gt;.&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;