The Riemann zeta function and tuning: Difference between revisions

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where the product is over all primes ''p''. The product converges for values of ''s'' with real part greater than one, while at {{nowrap|''s'' {{=}} 1}} it diverges to infinity. We may remove a finite list of primes from consideration by multiplying ζ(''s'') by the corresponding factors {{nowrap|(1 − ''p''<sup>−''s''</sup>)}} 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, {{nowrap|(1 − 2<sup>−''s''</sup>)ζ(''s'')}} is now dominated by 3, and the large peak values occur near equal divisions of the "tritave", ie 3.

Along the critical line:

Along any line of constant <math>\sigma</math>, [[The Riemann zeta function and tuning/Appendix#1b. Conversion factor for removing primes|it can be shown that]]:

<math>\displaystyle{

\left| 1 - p^{-\sigma - it} \right| = \sqrt{1 + \frac{1}{p^{2\sigma}} - \frac{2 \cos(t \ln p)}{p^\sigma}}

}</math>;

in particular, on the critical line:

<math>\displaystyle{

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Line 700:

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.

For example, ~~the function used~~ to find peak ~~values of EDTs~~ (division of the [[3/1|{{ordinal|3}}]] harmonic, or "tritave") is

For example, if we want to find zeta peak [[EDT]]s (division of the [[3/1|{{ordinal|3}}]] harmonic, or "tritave") - noting that here we must substitute <math>t = \frac{2\pi x}{ln(3)}</math> instead of <math>\frac{2\pi x}{ln(2)}</math> - in the no-twos subgroup, our modified Z function is:

<math>

\displaystyle Z\left(\frac{2\pi}{\ln(3)}t\right)\sqrt{\frac{3}{2}-\sqrt{2}\cos\left(\frac{2\pi\ln(2)}{\ln(3)}t\right)}

\displaystyle Z\left(\frac{2\pi}{\ln(3)}x\right)\sqrt{\frac{3}{2}-\sqrt{2}\cos\left(\frac{2\pi\ln(2)}{\ln(3)}x\right)}

</math>

</math>.

Removing 2 leads to increasing adjusted peak values corresponding to edts into {{EDTs| 4, 7, 9, 13, 15, 17, 26, 32, 39, 56, 69, 75, 88, 131, 245, 316,…}} parts. We can also compare zeta peak EDTs with pure and tempered tritaves just like [[#zeta peak edos|zeta peak]] edos. A striking feature of this list is the appearance not only of [[13edt]], the [[Bohlen–Pierce]] division of the tritave, but the multiples 26, 39~~, and 52~~ also.

Removing 2 leads to increasing adjusted peak values corresponding to edts into {{EDTs| 4, 7, 9, 13, 15, 17, 26, 32, 39, 56, 69, 75, 88, 131, 245, 316,…}} parts. We can also compare zeta peak EDTs with pure and tempered tritaves just like [[#zeta peak edos|zeta peak]] edos. A striking feature of this list is the appearance not only of [[13edt]], the [[Bohlen–Pierce]] division of the tritave, but the multiples 26 and 39 also.

== Open problems ==

@@ Line 686: / Line 686: @@
 where the product is over all primes ''p''. The product converges for values of ''s'' with real part greater than one, while at {{nowrap|''s'' {{=}} 1}} it diverges to infinity. We may remove a finite list of primes from consideration by multiplying ζ(''s'') by the corresponding factors {{nowrap|(1 − ''p''<sup>−''s''</sup>)}} 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, {{nowrap|(1 − 2<sup>−''s''</sup>)ζ(''s'')}} is now dominated by 3, and the large peak values occur near equal divisions of the "tritave", ie 3.
-Along the critical line:
+Along any line of constant <math>\sigma</math>, [[The Riemann zeta function and tuning/Appendix#1b. Conversion factor for removing primes|it can be shown that]]:
+<math>\displaystyle{
+\left| 1 - p^{-\sigma - it} \right| = \sqrt{1 + \frac{1}{p^{2\sigma}} - \frac{2 \cos(t \ln p)}{p^\sigma}}
+}</math>;
+in particular, on the critical line:
 <math>\displaystyle{
@@ Line 694: / Line 700: @@
 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.
-For example, the function used to find peak values of EDTs (division of the [[3/1|{{ordinal|3}}]] harmonic, or "tritave") is
+For example, if we want to find zeta peak [[EDT]]s (division of the [[3/1|{{ordinal|3}}]] harmonic, or "tritave") - noting that here we must substitute <math>t = \frac{2\pi x}{ln(3)}</math> instead of <math>\frac{2\pi x}{ln(2)}</math> - in the no-twos subgroup, our modified Z function is:
 <math>
-\displaystyle Z\left(\frac{2\pi}{\ln(3)}t\right)\sqrt{\frac{3}{2}-\sqrt{2}\cos\left(\frac{2\pi\ln(2)}{\ln(3)}t\right)}
+\displaystyle Z\left(\frac{2\pi}{\ln(3)}x\right)\sqrt{\frac{3}{2}-\sqrt{2}\cos\left(\frac{2\pi\ln(2)}{\ln(3)}x\right)}
-</math>
+</math>.
-Removing 2 leads to increasing adjusted peak values corresponding to edts into {{EDTs| 4, 7, 9, 13, 15, 17, 26, 32, 39, 56, 69, 75, 88, 131, 245, 316,…}} parts. We can also compare zeta peak EDTs with pure and tempered tritaves just like [[#zeta peak edos|zeta peak]] edos. A striking feature of this list is the appearance not only of [[13edt]], the [[Bohlen–Pierce]] division of the tritave, but the multiples 26, 39, and 52 also.
+Removing 2 leads to increasing adjusted peak values corresponding to edts into {{EDTs| 4, 7, 9, 13, 15, 17, 26, 32, 39, 56, 69, 75, 88, 131, 245, 316,…}} parts. We can also compare zeta peak EDTs with pure and tempered tritaves just like [[#zeta peak edos|zeta peak]] edos. A striking feature of this list is the appearance not only of [[13edt]], the [[Bohlen–Pierce]] division of the tritave, but the multiples 26 and 39 also.
 == Open problems ==