The Riemann zeta function and tuning: Difference between revisions

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== Into the critical strip ==

So long as s is greater than or equal to one, the absolute value of the zeta function can be seen as a relative error measurement. However, the rationale for that view of things departs when s is less than one, particularly in the [http://mathworld.wolfram.com/CriticalStrip.html critical strip], when s lies between zero and one. As s approaches the value s=1/2 of the [http://mathworld.wolfram.com/CriticalLine.html critical line], the information content, so to speak, of the zeta function concerning higher primes increases and it behaves increasingly like a badness measure (or more correctly, since we have inverted it, like a goodness measure.) The quasi-symmetric [~~http~~://planetmath.org/encyclopedia/FunctionalEquationOfTheRiemannZetaFunction.html functional equation] of the zeta function tells us that past the critical line the information content starts to decrease again, with 1-s and s having the same information content. Hence it is the zeta function between s=1/2 and s=1, and especially the zeta function along the critical line s=1/2, which is of the most interest.

So long as s is greater than or equal to one, the absolute value of the zeta function can be seen as a relative error measurement. However, the rationale for that view of things departs when s is less than one, particularly in the [http://mathworld.wolfram.com/CriticalStrip.html critical strip], when s lies between zero and one. As s approaches the value s=1/2 of the [http://mathworld.wolfram.com/CriticalLine.html critical line], the information content, so to speak, of the zeta function concerning higher primes increases and it behaves increasingly like a badness measure (or more correctly, since we have inverted it, like a goodness measure.) The quasi-symmetric [https://planetmath.org/encyclopedia/FunctionalEquationOfTheRiemannZetaFunction.html functional equation] of the zeta function tells us that past the critical line the information content starts to decrease again, with 1-s and s having the same information content. Hence it is the zeta function between s=1/2 and s=1, and especially the zeta function along the critical line s=1/2, which is of the most interest.

As s>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 >> 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 s = +∞ with x an integer, we can trace a line back towards the critical strip on which zeta is real. Since when s >> 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 ζ(1/2 + i g) are real are called "Gram points", after [[Wikipedia:Jørgen 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.

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Note that for one of its neighbors, 271, it isn't entirely clear which peak value corresponds to the line of real values from +∞. This can be determined by looking at the absolute value of zeta along other s values, such as s=1 or s=3/4, and in this case the local minimum at 271.069 is the value in question. However, other peak values are not without their interest; the local maximum at 270.941, for instance, is associated to a different mapping for 3.

To generate this plot using the free version of Wolfram Cloud, you can copy-paste '''Plot[RiemannSiegelZ[9.06472028x], {x, 11.9,12.1}]''' and then in the menu select '''Evaluation > Evaluate Cells'''.

= Mike Battaglia's Expanded Results =

@@ Line 48: / Line 48: @@
 == Into the critical strip ==
-So long as s is greater than or equal to one, the absolute value of the zeta function can be seen as a relative error measurement. However, the rationale for that view of things departs when s is less than one, particularly in the [http://mathworld.wolfram.com/CriticalStrip.html critical strip], when s lies between zero and one. As s approaches the value s=1/2 of the [http://mathworld.wolfram.com/CriticalLine.html critical line], the information content, so to speak, of the zeta function concerning higher primes increases and it behaves increasingly like a badness measure (or more correctly, since we have inverted it, like a goodness measure.) The quasi-symmetric [http://planetmath.org/encyclopedia/FunctionalEquationOfTheRiemannZetaFunction.html functional equation] of the zeta function tells us that past the critical line the information content starts to decrease again, with 1-s and s having the same information content. Hence it is the zeta function between s=1/2 and s=1, and especially the zeta function along the critical line s=1/2, which is of the most interest.
+So long as s is greater than or equal to one, the absolute value of the zeta function can be seen as a relative error measurement. However, the rationale for that view of things departs when s is less than one, particularly in the [http://mathworld.wolfram.com/CriticalStrip.html critical strip], when s lies between zero and one. As s approaches the value s=1/2 of the [http://mathworld.wolfram.com/CriticalLine.html critical line], the information content, so to speak, of the zeta function concerning higher primes increases and it behaves increasingly like a badness measure (or more correctly, since we have inverted it, like a goodness measure.) The quasi-symmetric [https://planetmath.org/encyclopedia/FunctionalEquationOfTheRiemannZetaFunction.html functional equation] of the zeta function tells us that past the critical line the information content starts to decrease again, with 1-s and s having the same information content. Hence it is the zeta function between s=1/2 and s=1, and especially the zeta function along the critical line s=1/2, which is of the most interest.
 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 s = +∞ 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 ζ(1/2 + i g) are real are called "Gram points", after [[Wikipedia:Jørgen 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.
@@ Line 92: / Line 92: @@
 Note that for one of its neighbors, 271, it isn't entirely clear which peak value corresponds to the line of real values from +∞. This can be determined by looking at the absolute value of zeta along other s values, such as s=1 or s=3/4, and in this case the local minimum at 271.069 is the value in question. However, other peak values are not without their interest; the local maximum at 270.941, for instance, is associated to a different mapping for 3.
+To generate this plot using the free version of Wolfram Cloud, you can copy-paste '''Plot[RiemannSiegelZ[9.06472028x], {x, 11.9,12.1}]''' and then in the menu select '''Evaluation > Evaluate Cells'''.
 = Mike Battaglia's Expanded Results =