Riemann zeta function: Difference between revisions
ArrowHead294 (talk | contribs) mNo edit summary |
ArrowHead294 (talk | contribs) m Hex entities instead of decimal |
||
| Line 11: | Line 11: | ||
Suppose ''x'' is a variable representing some equal division of the octave. For example, if {{nowrap|''x'' {{=}} 80}}, ''x'' reflects 80edo with a step size of 15 cents and with pure octaves. Suppose that ''x'' can also be continuous, so that it can also represent fractional or "nonoctave" divisions as well. The [[Bohlen-Pierce|Bohlen-Pierce scale]], 13 equal divisions of 3/1, is approximately 8.202 equal divisions of the "octave" (although the octave itself does not appear in this tuning), and would hence be represented by a value of {{nowrap|''x'' {{=}} 8.202}}. | Suppose ''x'' is a variable representing some equal division of the octave. For example, if {{nowrap|''x'' {{=}} 80}}, ''x'' reflects 80edo with a step size of 15 cents and with pure octaves. Suppose that ''x'' can also be continuous, so that it can also represent fractional or "nonoctave" divisions as well. The [[Bohlen-Pierce|Bohlen-Pierce scale]], 13 equal divisions of 3/1, is approximately 8.202 equal divisions of the "octave" (although the octave itself does not appear in this tuning), and would hence be represented by a value of {{nowrap|''x'' {{=}} 8.202}}. | ||
Now suppose that & | Now suppose that ⌊''x''⌉ denotes the difference between ''x'' and the integer nearest to ''x'': | ||
<math>\rround{x} = \abs{x - \floor{x + \frac{1}{2}}}</math> | <math>\rround{x} = \abs{x - \floor{x + \frac{1}{2}}}</math> | ||
For example, & | For example, ⌊8.202⌉ would be 0.202, since it is the difference between 8.202 and the nearest integer, which is 8. Meanwhile, ⌊7.95⌉ would be 0.05, which is the difference between 7.95 and the nearest integer, which is 8. | ||
For any value of ''x'', we can construct a ''p''-limit [[Patent_val|generalized patent val]]. We do so by rounding {{nowrap|''x'' log<sub>2</sub>(''q'')}} to the nearest integer for each prime ''q'' up to ''p''. Now consider the function | For any value of ''x'', we can construct a ''p''-limit [[Patent_val|generalized patent val]]. We do so by rounding {{nowrap|''x'' log<sub>2</sub>(''q'')}} to the nearest integer for each prime ''q'' up to ''p''. Now consider the function | ||
| Line 33: | Line 33: | ||
where the summation is taken formally over all positive integers, though only the primes and prime powers make a nonzero contribution. | where the summation is taken formally over all positive integers, though only the primes and prime powers make a nonzero contribution. | ||
Another consequence of the above definition which might be objected to is that it results in a function with a {{w|Continuous function#Relation to differentiability and integrability|discontinuous derivative}}, whereas a smooth function be preferred. The function & | Another consequence of the above definition which might be objected to is that it results in a function with a {{w|Continuous function#Relation to differentiability and integrability|discontinuous derivative}}, whereas a smooth function be preferred. The function ⌊''x''⌉<sup>2</sup> is quadratically increasing near integer values of ''x'', and is periodic with period 1. Another function with these same properties is {{nowrap|1 − cos(2π''x'')}}, which is a smooth and in fact an {{w|entire function}}. Let us therefore now define for any {{nowrap|''s'' > 1}}: | ||
<math>\displaystyle E_s(x) = \sum_{n \geq 1} \frac{\Lambda(n)}{\ln n} \frac{1 - \cos(2 \pi x \log_2 n)}{n^s}</math> | <math>\displaystyle E_s(x) = \sum_{n \geq 1} \frac{\Lambda(n)}{\ln n} \frac{1 - \cos(2 \pi x \log_2 n)}{n^s}</math> | ||
| Line 56: | Line 56: | ||
So long as {{nowrap|''s'' ≥ 1}}, 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 {{nowrap|''s'' < 1}}, particularly in the [http://mathworld.wolfram.com/CriticalStrip.html critical strip], when {{nowrap|0 < ''s'' < 1}}. As s approaches the value {{nowrap|''s'' {{=}} {{sfrac|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 {{nowrap|1 − ''s''}} and ''s'' having the same information content. Hence it is the zeta function between {{nowrap|''s'' {{=}} {{sfrac|1|2}}}} and {{nowrap|''s'' {{=}} 1}}, and especially the zeta function along the critical line {{nowrap|''s'' {{=}} {{sfrac|1|2}}}}, which is of the most interest. | So long as {{nowrap|''s'' ≥ 1}}, 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 {{nowrap|''s'' < 1}}, particularly in the [http://mathworld.wolfram.com/CriticalStrip.html critical strip], when {{nowrap|0 < ''s'' < 1}}. As s approaches the value {{nowrap|''s'' {{=}} {{sfrac|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 {{nowrap|1 − ''s''}} and ''s'' having the same information content. Hence it is the zeta function between {{nowrap|''s'' {{=}} {{sfrac|1|2}}}} and {{nowrap|''s'' {{=}} 1}}, and especially the zeta function along the critical line {{nowrap|''s'' {{=}} {{sfrac|1|2}}}}, which is of the most interest. | ||
As {{nowrap|''s'' > 1}} gets larger, the Dirichlet series for the zeta function is increasingly dominated by the 2 term, getting ever closer to simply {{nowrap|1 + 2<sup>−''z''</sup>}}, which approaches 1 as {{nowrap|''s'' {{=}} Re(''z'')}} becomes larger. When {{nowrap|''s'' &# | As {{nowrap|''s'' > 1}} gets larger, the Dirichlet series for the zeta function is increasingly dominated by the 2 term, getting ever closer to simply {{nowrap|1 + 2<sup>−''z''</sup>}}, which approaches 1 as {{nowrap|''s'' {{=}} Re(''z'')}} becomes larger. When {{nowrap|''s'' ≫ 1}} and ''x'' is an integer, the real part of zeta is approximately {{nowrap|1 + 2<sup>−''s''</sup>}}, and the imaginary part is approximately zero; that is, zeta is approximately real. Starting from {{nowrap|''s'' {{=}} +∞}} with ''x'' an integer, we can trace a line back towards the critical strip on which zeta is real. Since when {{nowrap|''s'' ≫ 1}} the derivative is approximately −{{sfrac|ln(2)|2<sup>''s''</sup>}}, 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 {{nowrap|''s'' {{=}} {{sfrac|1|2}}}}, it produces a real value of zeta on the critical line. Points on the critical line where {{nowrap|ζ({{frac|1|2}} + ''ig'')}} are real are called "Gram points", after {{w|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 +∞ 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 −ζ{{'}}(''z'') is getting a boost from other small primes as it travels toward the Gram point. A complex formula due to {{w|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 {{w|Riemann–Siegel formula}} since {{w|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 {{nowrap|ζ({{frac|1|2}} + ''ig'')}} at the corresponding Gram point should be especially large. | Because the value of zeta increased continuously as it made its way from +∞ 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 −ζ{{'}}(''z'') is getting a boost from other small primes as it travels toward the Gram point. A complex formula due to {{w|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 {{w|Riemann–Siegel formula}} since {{w|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 {{nowrap|ζ({{frac|1|2}} + ''ig'')}} at the corresponding Gram point should be especially large. | ||
| Line 116: | Line 116: | ||
\zeta(s) = \sum_n n^{-s}</math> | \zeta(s) = \sum_n n^{-s}</math> | ||
Now let's do two things: we're going to expand {{nowrap|''s'' {{=}} σ + ''it''}}, and we're going to multiply ζ(s) by its conjugate ζ(''s''){{'}}, noting that {{nowrap|ζ(''s''){{'}} {{=}} ζ(''s''{{'}})}} and {{nowrap|ζ(''s'') &# | Now let's do two things: we're going to expand {{nowrap|''s'' {{=}} σ + ''it''}}, and we're going to multiply ζ(s) by its conjugate ζ(''s''){{'}}, noting that {{nowrap|ζ(''s''){{'}} {{=}} ζ(''s''{{'}})}} and {{nowrap|ζ(''s'') ⋅ ζ(''s''){{'}} {{=}} ζ(''s'')<sup>2</sup>}}. We get: | ||
<math> \displaystyle | <math> \displaystyle | ||
| Line 189: | Line 189: | ||
For every strictly positive rational ''n''/''d'', there is a cosine with period {{nowrap|2π log<sub>2</sub>({{frac|''n''|''d''}})}}. This cosine peaks at {{nowrap|''x'' {{=}} {{sfrac|''N''|log<sub>2</sub>(''n''/''d'')}}}} for all integers ''N'', or in other words, the Nth-equal division of the rational number {{frac|''n''|''d''}}, and hits troughs midway between. | For every strictly positive rational ''n''/''d'', there is a cosine with period {{nowrap|2π log<sub>2</sub>({{frac|''n''|''d''}})}}. This cosine peaks at {{nowrap|''x'' {{=}} {{sfrac|''N''|log<sub>2</sub>(''n''/''d'')}}}} for all integers ''N'', or in other words, the Nth-equal division of the rational number {{frac|''n''|''d''}}, and hits troughs midway between. | ||
Our mysterious substitution above was chosen to set the units for this up nicely. The variable x now happens to be measured in divisions of the octave. (The original variable ''t'', which was the imaginary part of the zeta argument ''s'', can be thought of as the number of divisions of the interval {{nowrap|''e''<sup>2π</sup> &# | Our mysterious substitution above was chosen to set the units for this up nicely. The variable x now happens to be measured in divisions of the octave. (The original variable ''t'', which was the imaginary part of the zeta argument ''s'', can be thought of as the number of divisions of the interval {{nowrap|''e''<sup>2π</sup> ≈ 535.49}}, or what [[Keenan_Pepper|Keenan Pepper]] has called the "natural interval.") | ||
As mentioned in Gene's original zeta derivation, these cosine functions can be thought of as good approximations to the terms in the TE error computation, which are all the squared errors for the different primes. Rather than taking the square of the error, we instead put the error through the function {{sfrac|1 − cos(''x'')|2}}, which is "close enough" for small values of ''x''. Since we are always rounding off to the best mapping, this error is never more 0.5 steps of the EDO, so since we have {{nowrap|−0.5 < x < 0.5}} we have a decent enough approximation. | As mentioned in Gene's original zeta derivation, these cosine functions can be thought of as good approximations to the terms in the TE error computation, which are all the squared errors for the different primes. Rather than taking the square of the error, we instead put the error through the function {{sfrac|1 − cos(''x'')|2}}, which is "close enough" for small values of ''x''. Since we are always rounding off to the best mapping, this error is never more 0.5 steps of the EDO, so since we have {{nowrap|−0.5 < x < 0.5}} we have a decent enough approximation. | ||
| Line 351: | Line 351: | ||
|- | |- | ||
! rowspan="2" | Zeta<br />peak !! Detuned<br />octaves | ! rowspan="2" | Zeta<br />peak !! Detuned<br />octaves | ||
| &# | | ★ | ||
| &# | | ★ | ||
| &# | | ★ | ||
| &# | | ★ | ||
| &# | | ★ | ||
| &# | | ★ | ||
| &# | | ★ | ||
| &# | | ★ | ||
| &# | | ★ | ||
| &# | | ★ | ||
| &# | | ★ | ||
| &# | | ★ | ||
| &# | | ★ | ||
| | | | ||
| &# | | ★ | ||
| &# | | ★ | ||
| | | | ||
| &# | | ★ | ||
| &# | | ★ | ||
| &# | | ★ | ||
| &# | | ★ | ||
| &# | | ★ | ||
| &# | | ★ | ||
| &# | | ★ | ||
| &# | | ★ | ||
| | | | ||
| &# | | ★ | ||
| &# | | ★ | ||
| &# | | ★ | ||
| | | | ||
| &# | | ★ | ||
| &# | | ★ | ||
| &# | | ★ | ||
| &# | | ★ | ||
| &# | | ★ | ||
|- | |- | ||
! Pure<br />octaves | ! Pure<br />octaves | ||
| &# | | ★ | ||
| &# | | ★ | ||
| &# | | ★ | ||
| | | | ||
| &# | | ★ | ||
| &# | | ★ | ||
| &# | | ★ | ||
| &# | | ★ | ||
| &# | | ★ | ||
| &# | | ★ | ||
| | | | ||
| &# | | ★ | ||
| &# | | ★ | ||
| | | | ||
| &# | | ★ | ||
| | | | ||
| &# | | ★ | ||
| | | | ||
| &# | | ★ | ||
| &# | | ★ | ||
| | | | ||
| &# | | ★ | ||
| | | | ||
| &# | | ★ | ||
| &# | | ★ | ||
| &# | | ★ | ||
| | | | ||
| | | | ||
| | | | ||
| &# | | ★ | ||
| &# | | ★ | ||
| | | | ||
| | | | ||
| Line 426: | Line 426: | ||
! colspan="2" | Zeta integral | ! colspan="2" | Zeta integral | ||
| | | | ||
| &# | | ★ | ||
| | | | ||
| | | | ||
| &# | | ★ | ||
| &# | | ★ | ||
| | | | ||
| &# | | ★ | ||
| &# | | ★ | ||
| | | | ||
| | | | ||
| &# | | ★ | ||
| &# | | ★ | ||
| | | | ||
| &# | | ★ | ||
| &# | | ★ | ||
| | | | ||
| | | | ||
| | | | ||
| &# | | ★ | ||
| | | | ||
| &# | | ★ | ||
| | | | ||
| &# | | ★ | ||
| &# | | ★ | ||
| | | | ||
| | | | ||
| Line 457: | Line 457: | ||
| | | | ||
| | | | ||
| &# | | ★ | ||
| | | | ||
| &# | | ★ | ||
|- | |- | ||
! colspan="2" | Zeta gap | ! colspan="2" | Zeta gap | ||
| | | | ||
| &# | | ★ | ||
| &# | | ★ | ||
| | | | ||
| &# | | ★ | ||
| &# | | ★ | ||
| | | | ||
| &# | | ★ | ||
| &# | | ★ | ||
| | | | ||
| | | | ||
| &# | | ★ | ||
| | | | ||
| &# | | ★ | ||
| &# | | ★ | ||
| &# | | ★ | ||
| | | | ||
| | | | ||
| Line 486: | Line 486: | ||
| | | | ||
| | | | ||
| &# | | ★ | ||
| &# | | ★ | ||
| | | | ||
| | | | ||
| Line 496: | Line 496: | ||
| | | | ||
| | | | ||
| &# | | ★ | ||
|} | |} | ||
| Line 645: | Line 645: | ||
}</math> | }</math> | ||
From this we may deduce that {{nowrap|{{sfrac|θ(''t'')|π}} &# | From this we may deduce that {{nowrap|{{sfrac|θ(''t'')|π}} ≈ ''r'' ln(''r'') − ''r'' − {{sfrac|1|8}}}}, where {{nowrap|''r'' {{=}} {{sfrac|''t''|2π}} {{=}} {{sfrac|''x'' ln(2)}}}}; hence while x is the number of equal steps to an octave, ''r'' is the number of equal steps to an "''e''-tave", meaning the interval of ''e'', which is {{nowrap|{{sfrac|1200|ln(2)}} {{=}} 1731.234{{c}}}}. | ||
Recall that Gram points near to pure-octave edos, where ''x'' is an integer, can be expected to correspond to peak values of {{nowrap|{{!}}ζ{{!}} {{=}} {{!}}Z{{!}}}}. We can find these Gram points by Newton's method applied to the above formula. If {{nowrap|''r'' {{=}} {{sfrac|''x''|ln(2)}}}}, and if {{nowrap|''n'' {{=}} ⌊''r'' ln(''r'') − ''r'' + {{frac|3|8}}⌋}} is the nearest integer to {{sfrac|θ(2π''r'')|π}}, then we may set {{nowrap|''r''<sup>+</sup> {{=}} {{sfrac|''r'' + ''n'' + {{sfrac|1|8}}|ln(''r'')}}}}. This is the first iteration of Newton's method, which we may repeat if we like, but in fact no more than one iteration is really required. This is the first black magic formula, giving an adjusted "Gram" tuning from the orginal one. | Recall that Gram points near to pure-octave edos, where ''x'' is an integer, can be expected to correspond to peak values of {{nowrap|{{!}}ζ{{!}} {{=}} {{!}}Z{{!}}}}. We can find these Gram points by Newton's method applied to the above formula. If {{nowrap|''r'' {{=}} {{sfrac|''x''|ln(2)}}}}, and if {{nowrap|''n'' {{=}} ⌊''r'' ln(''r'') − ''r'' + {{frac|3|8}}⌋}} is the nearest integer to {{sfrac|θ(2π''r'')|π}}, then we may set {{nowrap|''r''<sup>+</sup> {{=}} {{sfrac|''r'' + ''n'' + {{sfrac|1|8}}|ln(''r'')}}}}. This is the first iteration of Newton's method, which we may repeat if we like, but in fact no more than one iteration is really required. This is the first black magic formula, giving an adjusted "Gram" tuning from the orginal one. | ||
| Line 656: | Line 656: | ||
There are various approaches to the question of computing the zeta function, but perhaps the simplest is the use of the {{w|Dirichlet eta function}} which was introduced to mathematics by {{w|Johann Peter Gustav Lejeune Dirichlet}}, who despite his name was a German and the brother-in-law of {{w|Felix Mendelssohn}}. | There are various approaches to the question of computing the zeta function, but perhaps the simplest is the use of the {{w|Dirichlet eta function}} which was introduced to mathematics by {{w|Johann Peter Gustav Lejeune Dirichlet}}, who despite his name was a German and the brother-in-law of {{w|Felix Mendelssohn}}. | ||
The zeta function has a [http://mathworld.wolfram.com/SimplePole.html simple pole] at {{nowrap|''z'' {{=}} 1}} which forms a barrier against continuing it with its {{w|Euler product}} or {{w|Dirichlet series}} representation. We could subtract off the pole, or multiply by a factor of {{nowrap|''z'' − 1}}, but at the expense of losing the character of a Dirichlet series or Euler product. A better method is to multiply by a factor of {{nowrap|1 − 2<sup>1&# | The zeta function has a [http://mathworld.wolfram.com/SimplePole.html simple pole] at {{nowrap|''z'' {{=}} 1}} which forms a barrier against continuing it with its {{w|Euler product}} or {{w|Dirichlet series}} representation. We could subtract off the pole, or multiply by a factor of {{nowrap|''z'' − 1}}, but at the expense of losing the character of a Dirichlet series or Euler product. A better method is to multiply by a factor of {{nowrap|1 − 2<sup>1 − ''z''</sup>}}, leading to the eta function: | ||
<math>\displaystyle{\eta(z) = \left(1-2^{1-z}\right)\zeta(z) = \sum_{n=1}^\infty (-1)^{n-1} n^{-z} | <math>\displaystyle{\eta(z) = \left(1-2^{1-z}\right)\zeta(z) = \sum_{n=1}^\infty (-1)^{n-1} n^{-z} | ||