Xenharmonic Wiki

Riemann zeta function: Difference between revisions

← Older editNewer edit →
VisualWikitext
Sintel (talk | contribs)
autopatrolled, emailconfirmed
1,636 edits
→Mike Battaglia's expanded results: minor fixes and notation consistency
Sintel (talk | contribs)
autopatrolled, emailconfirmed
1,636 edits
Line 48: Line 48:
Now suppose that [https://www.desmos.com/calculator/krigk43int {{rr|''x''}}] denotes the difference between ''x'' and the integer nearest to ''x'':
Now suppose that [https://www.desmos.com/calculator/krigk43int {{rr|''x''}}] denotes the difference between ''x'' and the integer nearest to ''x'':


<math>\rround{x} = \abs{x - \floor{x + \frac{1}{2}}}</math>
<math>
\lfloor x \rceil = \left| x - \left\lfloor x + \frac{1}{2} \right\rfloor \right|
</math>


For example, {{nowrap|{{rr|8.202}} {{=}} 0.202}}, since it is the difference between 8.202 and the nearest integer, which is 8. Meanwhile, {{nowrap|{{rr|7.95}} {{=}} 0.05}}, which is the difference between 7.95 and the nearest integer, which is 8. This represents the absolute relative error of the octave in equal tuning ''x'', or alternatively how much x is detuned from an edo.
For example, {{nowrap|{{rr|8.202}} {{=}} 0.202}}, since it is the difference between 8.202 and the nearest integer, which is 8. Meanwhile, {{nowrap|{{rr|7.95}} {{=}} 0.05}}, which is the difference between 7.95 and the nearest integer, which is 8. This represents the absolute relative error of the octave in equal tuning ''x'', or alternatively how much x is detuned from an edo.
Line 54: Line 56:
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''. For example, for {{nowrap|''x'' {{=}} 12}}, we find 2 at 12, 3 at 19, 5 at 28, etc. Now consider [https://www.desmos.com/calculator/4uamhon9tt 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''. For example, for {{nowrap|''x'' {{=}} 12}}, we find 2 at 12, 3 at 19, 5 at 28, etc. Now consider [https://www.desmos.com/calculator/4uamhon9tt the function]


<math>\displaystyle \xi_p(x) = \sum_{\substack{2 \leq q \leq p \\ q \text{ prime}}} \left(\frac{\rround{x \log_2 q}}{\log_2 q}\right)^2</math>
<math>
\displaystyle \xi_p(x)
= \sum_{\substack{2 \leq q \leq p \\ q \text{ prime}}} \left(\frac{\lfloor x \log_2 q \rceil}{\log_2 q}\right)^2
</math>


Here, the numerator represents the relative error on each prime, and the denominator represents a weight factor of the logarithm of each prime, so the function represents a p-limit badness metric.
Here, the numerator represents the relative error on each prime, and the denominator represents a weight factor of the logarithm of each prime, so the function represents a p-limit badness metric.
Line 62: Line 67:
Now suppose we don't want a formula for any specific prime limit, but which applies to all primes. We can't take the above sum to infinity, since it doesn't converge. However, we could [https://www.desmos.com/calculator/0qhhewlsaz change the weighting factor to a power] so that it does converge:
Now suppose we don't want a formula for any specific prime limit, but which applies to all primes. We can't take the above sum to infinity, since it doesn't converge. However, we could [https://www.desmos.com/calculator/0qhhewlsaz change the weighting factor to a power] so that it does converge:


<math>\displaystyle \xi_\infty(x) = \sum_{\substack{q \geq 2 \\ q \text{ prime}}} \frac{\rround{x \log_2 q}^2}{q^s}</math>
<math>\displaystyle \xi_\infty(x) = \sum_{\substack{q \geq 2 \\ q \text{ prime}}} \frac{\lfloor x \log_2 q \rceil^2}{q^s}</math>


Seeing that we notate the power as ''s'', it might become apparent where the Riemann zeta function will eventually show up.
Seeing that we notate the power as ''s'', it might become apparent where the Riemann zeta function will eventually show up.
Line 68: Line 73:
If ''s'' is greater than one, this does converge. However, we might want to make a few adjustments. For one thing, if the error is low enough that the tuning is consistent, then the error of the square of a prime is twice that of the prime, of the cube tripled, and so forth until the error becomes inconsistent. When the weighting uses logarithms and error measures are consistent, then the logarithmic weighting cancels this effect out, so we might consider that prime powers were implicitly included in the Tenney-Euclidean measure. We can go ahead and include them by adding a factor of {{sfrac|1|''n''}} for each prime power ''p''<sup>''n''</sup>. A somewhat peculiar but useful way to write the result of doing this is in terms of the {{w|Von Mangoldt function}}, an {{w|arithmetic function}} on positive integers which is equal to ln(''p'') on prime powers ''p''<sup>''n''</sup>, and is zero elsewhere. This is written using a capital lambda, as Λ(''n''), and in terms of it we can include prime powers in our error function as
If ''s'' is greater than one, this does converge. However, we might want to make a few adjustments. For one thing, if the error is low enough that the tuning is consistent, then the error of the square of a prime is twice that of the prime, of the cube tripled, and so forth until the error becomes inconsistent. When the weighting uses logarithms and error measures are consistent, then the logarithmic weighting cancels this effect out, so we might consider that prime powers were implicitly included in the Tenney-Euclidean measure. We can go ahead and include them by adding a factor of {{sfrac|1|''n''}} for each prime power ''p''<sup>''n''</sup>. A somewhat peculiar but useful way to write the result of doing this is in terms of the {{w|Von Mangoldt function}}, an {{w|arithmetic function}} on positive integers which is equal to ln(''p'') on prime powers ''p''<sup>''n''</sup>, and is zero elsewhere. This is written using a capital lambda, as Λ(''n''), and in terms of it we can include prime powers in our error function as


<math>\displaystyle \xi_\infty(x) = \sum_{n \geq 1} \frac{\Lambda(n)}{\ln n} \frac{\rround{x \log_2 n}^2}{n^s}</math>
<math>\displaystyle \xi_\infty(x) = \sum_{n \geq 1} \frac{\Lambda(n)}{\ln n} \frac{\lfloor x \log_2 n \rceil^2}{n^s}</math>


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.
Line 84: Line 89:
This new function has the property that {{nowrap|F<sub>s</sub>(''x'') {{=}} F<sub>s</sub>(0) − E<sub>s</sub>(''x'')}}, so that all we have done is flip the sign of E<sub>s</sub>(''x'') and offset it vertically. This now increases to a maximum value for low errors, rather than declining to a minimum. Of more interest is the fact that it is a known mathematical function, which can be expressed in terms of the real part of the logarithm of the {{w|Riemann zeta function}}:
This new function has the property that {{nowrap|F<sub>s</sub>(''x'') {{=}} F<sub>s</sub>(0) − E<sub>s</sub>(''x'')}}, so that all we have done is flip the sign of E<sub>s</sub>(''x'') and offset it vertically. This now increases to a maximum value for low errors, rather than declining to a minimum. Of more interest is the fact that it is a known mathematical function, which can be expressed in terms of the real part of the logarithm of the {{w|Riemann zeta function}}:


<math>\displaystyle F_s(x) = \Re \ln \zeta\left(s + \frac{2 \pi i}{\ln 2}x\right)</math>
<math>\displaystyle F_s(x) = \mathrm{Re} \left( \ln \zeta \left(s + \frac{2 \pi i}{\ln 2}x \right) \right)</math>


{{Todo|expand|inline=1|text=Make it clear how Fₛ(x) relates to the zeta function. Due to the sudden appearance of the natural logarithm and the imaginary unit i, this appears to have to do with complex exponentials (i.e. those found in the denominator of the terms of zeta when the input is complex), but it would be more approachable if the precise derivation was laid out here.}}
{{Todo|expand|inline=1|text=Make it clear how Fₛ(x) relates to the zeta function. Due to the sudden appearance of the natural logarithm and the imaginary unit i, this appears to have to do with complex exponentials (i.e. those found in the denominator of the terms of zeta when the input is complex), but it would be more approachable if the precise derivation was laid out here.}}
Line 90: Line 95:
If we take exponentials of both sides, then
If we take exponentials of both sides, then


<math>\displaystyle \exp(F_s(x)) = \abs{\zeta\left(s + \frac{2 \pi i}{\ln 2}x\right)}</math>
<math>\displaystyle \exp(F_s(x)) = \left| \zeta\left(s + \frac{2 \pi i}{\ln 2}x\right) \right|</math>


so that we see that the absolute value of the zeta function serves to measure the relative error of an equal division.
so that we see that the absolute value of the zeta function serves to measure the relative error of an equal division.
Line 127: Line 132:


Since θ is holomorphic on the strip with imaginary part between −{{sfrac|1|2}} and {{sfrac|1|2}}, so is Z. Since the exponential function has no zeros, the zeros of Z in this strip correspond one to one with the zeros of ζ in the critical strip. Since the exponential of an imaginary argument has absolute value 1, the absolute value of Z along the real axis is the same as the absolute value of ζ at the corresponding place on the critical line. And since theta was defined so as to give precisely this property, Z is a real even function of the real variable ''t''.
Since θ is holomorphic on the strip with imaginary part between −{{sfrac|1|2}} and {{sfrac|1|2}}, so is Z. Since the exponential function has no zeros, the zeros of Z in this strip correspond one to one with the zeros of ζ in the critical strip. Since the exponential of an imaginary argument has absolute value 1, the absolute value of Z along the real axis is the same as the absolute value of ζ at the corresponding place on the critical line. And since theta was defined so as to give precisely this property, Z is a real even function of the real variable ''t''.
=== Plots ===


Using the [http://functions.wolfram.com/webMathematica/FunctionPlotting.jsp?name=RiemannSiegelZ online plotter] we can plot Z in the regions corresponding to scale divisions, using the conversion factor {{nowrap|''t'' {{=}} {{sfrac|2π|ln(2)}}''x''}}, for ''x'' a number near or at an edo number. Hence, for instance, to plot 12 plot around 108.777, to plot 31 plot around 281.006, and so forth. An alternative plotter is the applet [http://web.viu.ca/pughg/RiemannZeta/RiemannZetaLong.html here].
Using the [http://functions.wolfram.com/webMathematica/FunctionPlotting.jsp?name=RiemannSiegelZ online plotter] we can plot Z in the regions corresponding to scale divisions, using the conversion factor {{nowrap|''t'' {{=}} {{sfrac|2π|ln(2)}}''x''}}, for ''x'' a number near or at an edo number. Hence, for instance, to plot 12 plot around 108.777, to plot 31 plot around 281.006, and so forth. An alternative plotter is the applet [http://web.viu.ca/pughg/RiemannZeta/RiemannZetaLong.html here].
Retrieved from "https://en.xen.wiki/w/Riemann_zeta_function"