Riemann zeta function: Difference between revisions

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== Graph links ==

A link to the graph of zeta can be found at [https://samuelj.li/complex-function-plotter/#abs(zeta(i*2*pi*real(z)%2Fln(2)%2Bimag(z))) Zeta in Samuelj Plotter]. (In the top left menu, make sure that "Enable Checkerboard" is unticked and "Invert Gradient" and "Continuous Gradient" are ticked.) The function has been reoriented to place EDO size along the horizontal axis and weight along the vertical axis, and also scaled by ~~<math>\tfrac~~{~~2\pi}~~{\ln(2)}~~</math>~~ to ensure that the real number line aligns with edos. One can see that with higher weights, the function approaches a cyclic function with a period of 1; this corresponds to the prime 2 dominating more and more extremely as other harmonics are weighted less with higher weights. You can see this easier by raising the entire expression to an absurdly high power, such as 100. Note, however, that this visualization is inaccurate beyond a couple hundred: around 146.5, 324.5 and 473.5, and in many cases after, there appear to be zeroes that are not on the critical line; this is an artifact of the way the function is approximated and is the ultimate reason why the Riemann hypothesis remains unsolved. These actually correspond to zeroes that are very close together but on the critical line.

A link to the graph of zeta can be found at [https://samuelj.li/complex-function-plotter/#abs(zeta(i*2*pi*real(z)%2Fln(2)%2Bimag(z))) Zeta in Samuelj Plotter]. (In the top left menu, make sure that "Enable Checkerboard" is unticked and "Invert Gradient" and "Continuous Gradient" are ticked.) The function has been reoriented to place EDO size along the horizontal axis and weight along the vertical axis, and also scaled by {{sfrac|2π|ln(2)}} to ensure that the real number line aligns with edos. One can see that with higher weights, the function approaches a cyclic function with a period of 1; this corresponds to the prime 2 dominating more and more extremely as other harmonics are weighted less with higher weights. You can see this easier by raising the entire expression to an absurdly high power, such as 100. Note, however, that this visualization is inaccurate beyond a couple hundred: around 146.5, 324.5 and 473.5, and in many cases after, there appear to be zeroes that are not on the critical line; this is an artifact of the way the function is approximated and is the ultimate reason why the Riemann hypothesis remains unsolved. These actually correspond to zeroes that are very close together but on the critical line.

You may also view the graph of zeta along the critical line on Desmos: [https://www.desmos.com/calculator/dstp7wnidf Zeta in Desmos]. This makes it easier to see peaks, but only works for {{nowrap|σ {{=}} {{sfrac|1|2}}}}.

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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].

If you have access to {{w|Mathematica}}, which has Z, zeta and theta as a part of its suite of initially defined functions, you can do even better. Below is a Mathematica-generated plot of Z({{~~frac~~|2π''x''|ln(2)}}) in the region around [[12edo]]:

If you have access to {{w|Mathematica}}, which has Z, zeta and theta as a part of its suite of initially defined functions, you can do even better. Below is a Mathematica-generated plot of Z{{pars|{{sfrac|2π''x''|ln(2)}}|1.8}} in the region around [[12edo]]:

[[File:plot12.png|alt=plot12.png|plot12.png]]

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We'll deal with each of these separately.

First, in the leftmost summation, we can see that {{nowrap|''n'' {{=}} ''d''}} implies {{nowrap|ln({{~~frac~~|''n''|''d''}}) {{=}} 0}}. Since {{nowrap|sin(0) {{=}} 0}}, the sin term in the numerator cancels out, yielding:

First, in the leftmost summation, we can see that {{nowrap|''n'' {{=}} ''d''}} implies {{nowrap|ln{{pars|{{sfrac|''n''|''d''}}|1.8}} {{=}} 0}}. Since {{nowrap|sin(0) {{=}} 0}}, the sin term in the numerator cancels out, yielding:

<math> \displaystyle

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\frac{\cos\left(t \ln\left({\tfrac{q}{p}}\right)\right) - i\sin\left(t \ln\left({\tfrac{q}{p}}\right)\right)}{(pq)^{\sigma}}</math>

Now, noting that {{nowrap|ln({{~~frac~~|''p''|''q''}}) {{=}} −ln({{~~frac~~|''q''|''p''}})}} and that sin is an odd function, we can see that the sin terms cancel out, leaving

Now, noting that {{nowrap|ln{{pars|{{sfrac|''p''|''q''}}|1.8}} {{=}} −ln{{pars|{{sfrac|''q''|''p''}}|1.8}}}} and that sin is an odd function, we can see that the sin terms cancel out, leaving

<math> \displaystyle

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=== Interpretation of results: "cosine relative error" ===

For every strictly positive rational ''n''/''d'', there is a cosine with period {{nowrap|log2({{~~frac~~|''n''|''d''}})}}. This cosine peaks at {{nowrap|''x'' {{=}} {{sfrac|''N''|log2(''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|log2{{pars|{{sfrac|''n''|''d''}}|1.8}}}}. This cosine peaks at {{nowrap|''x'' {{=}} {{sfrac|''N''|log2(''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''2π ≈ 535.49}}, or what [[Keenan_Pepper|Keenan Pepper]] has called the "[[zetave|natural interval]].")

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=== The Z function: a mathematically convenient version of zeta ===

The absolute value of {{nowrap|ζ({{~~frac~~|1|2}} + ''ig'')}} 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 {{nowrap|ζ({{~~frac~~|1|2}} + ''it'')}} with respect to ''t'' will be zero; however this does not mean its derivative there will be zero. In fact, the {{w|Riemann hypothesis}} is equivalent to the claim that all zeros of {{nowrap|ζ′(''s'' + ''it'')}} occur when {{nowrap|''s'' > {{sfrac|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 {{w|''Z'' function}}, which is defined (in terms of the {{subpage|appendix|u|s=Z function and Riemann-Siegel theta function|text=Riemann-Siegel theta function}}) as:

The absolute value of {{nowrap|ζ{{pars|{{sfrac|1|2}} + ''ig''}}|1.8}} 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 {{nowrap|ζ{{pars|{{sfrac|1|2}} + ''it''}}|1.8}} with respect to ''t'' will be zero; however this does not mean its derivative there will be zero. In fact, the {{w|Riemann hypothesis}} is equivalent to the claim that all zeros of {{nowrap|ζ′(''s'' + ''it'')}} occur when {{nowrap|''s'' > {{sfrac|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 {{w|''Z'' function}}, which is defined (in terms of the {{subpage|appendix|u|s=Z function and Riemann-Siegel theta function|text=Riemann-Siegel theta function}}) as:

<math>Z(t) = \exp(i \theta(t)) \zeta\left(\frac{1}{2} + it\right)</math>.

@@ Line 36: / Line 36: @@
 == Graph links ==
-A link to the graph of zeta can be found at [https://samuelj.li/complex-function-plotter/#abs(zeta(i*2*pi*real(z)%2Fln(2)%2Bimag(z))) Zeta in Samuelj Plotter]. (In the top left menu, make sure that "Enable Checkerboard" is unticked and "Invert Gradient" and "Continuous Gradient" are ticked.) The function has been reoriented to place EDO size along the horizontal axis and weight along the vertical axis, and also scaled by <math>\tfrac{2\pi}{\ln(2)}</math> to ensure that the real number line aligns with edos. One can see that with higher weights, the function approaches a cyclic function with a period of 1; this corresponds to the prime 2 dominating more and more extremely as other harmonics are weighted less with higher weights. You can see this easier by raising the entire expression to an absurdly high power, such as 100. Note, however, that this visualization is inaccurate beyond a couple hundred: around 146.5, 324.5 and 473.5, and in many cases after, there appear to be zeroes that are not on the critical line; this is an artifact of the way the function is approximated and is the ultimate reason why the Riemann hypothesis remains unsolved. These actually correspond to zeroes that are very close together but on the critical line.
+A link to the graph of zeta can be found at [https://samuelj.li/complex-function-plotter/#abs(zeta(i*2*pi*real(z)%2Fln(2)%2Bimag(z))) Zeta in Samuelj Plotter]. (In the top left menu, make sure that "Enable Checkerboard" is unticked and "Invert Gradient" and "Continuous Gradient" are ticked.) The function has been reoriented to place EDO size along the horizontal axis and weight along the vertical axis, and also scaled by {{sfrac|2π|ln(2)}} to ensure that the real number line aligns with edos. One can see that with higher weights, the function approaches a cyclic function with a period of 1; this corresponds to the prime 2 dominating more and more extremely as other harmonics are weighted less with higher weights. You can see this easier by raising the entire expression to an absurdly high power, such as 100. Note, however, that this visualization is inaccurate beyond a couple hundred: around 146.5, 324.5 and 473.5, and in many cases after, there appear to be zeroes that are not on the critical line; this is an artifact of the way the function is approximated and is the ultimate reason why the Riemann hypothesis remains unsolved. These actually correspond to zeroes that are very close together but on the critical line.
 You may also view the graph of zeta along the critical line on Desmos: [https://www.desmos.com/calculator/dstp7wnidf Zeta in Desmos]. This makes it easier to see peaks, but only works for {{nowrap|σ {{=}} {{sfrac|1|2}}}}.
@@ Line 45: / Line 45: @@
 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].
-If you have access to {{w|Mathematica}}, which has Z, zeta and theta as a part of its suite of initially defined functions, you can do even better. Below is a Mathematica-generated plot of Z({{frac|2π''x''|ln(2)}}) in the region around [[12edo]]:
+If you have access to {{w|Mathematica}}, which has Z, zeta and theta as a part of its suite of initially defined functions, you can do even better. Below is a Mathematica-generated plot of Z{{pars|{{sfrac|2π''x''|ln(2)}}|1.8}} in the region around [[12edo]]:
 [[File:plot12.png|alt=plot12.png|plot12.png]]
@@ Line 187: / Line 187: @@
 We'll deal with each of these separately.
-First, in the leftmost summation, we can see that {{nowrap|''n'' {{=}} ''d''}} implies {{nowrap|ln({{frac|''n''|''d''}}) {{=}} 0}}. Since {{nowrap|sin(0) {{=}} 0}}, the sin term in the numerator cancels out, yielding:
+First, in the leftmost summation, we can see that {{nowrap|''n'' {{=}} ''d''}} implies {{nowrap|ln{{pars|{{sfrac|''n''|''d''}}|1.8}} {{=}} 0}}. Since {{nowrap|sin(0) {{=}} 0}}, the sin term in the numerator cancels out, yielding:
 <math> \displaystyle
@@ Line 202: / Line 202: @@
 \frac{\cos\left(t \ln\left({\tfrac{q}{p}}\right)\right) - i\sin\left(t \ln\left({\tfrac{q}{p}}\right)\right)}{(pq)^{\sigma}}</math>
-Now, noting that {{nowrap|ln({{frac|''p''|''q''}}) {{=}} −ln({{frac|''q''|''p''}})}} and that sin is an odd function, we can see that the sin terms cancel out, leaving
+Now, noting that {{nowrap|ln{{pars|{{sfrac|''p''|''q''}}|1.8}} {{=}} −ln{{pars|{{sfrac|''q''|''p''}}|1.8}}}} and that sin is an odd function, we can see that the sin terms cancel out, leaving
 <math> \displaystyle
@@ Line 228: / Line 228: @@
 === Interpretation of results: "cosine relative error" ===
-For every strictly positive rational ''n''/''d'', there is a cosine with period {{nowrap|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|log<sub>2</sub>{{pars|{{sfrac|''n''|''d''}}|1.8}}}}. 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> ≈ 535.49}}, or what [[Keenan_Pepper|Keenan Pepper]] has called the "[[zetave|natural interval]].")
@@ Line 337: / Line 337: @@
 === The Z function: a mathematically convenient version of zeta ===
-The absolute value of {{nowrap|ζ({{frac|1|2}} + ''ig'')}} 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 {{nowrap|ζ({{frac|1|2}} + ''it'')}} with respect to ''t'' will be zero; however this does not mean its derivative there will be zero. In fact, the {{w|Riemann hypothesis}} is equivalent to the claim that all zeros of {{nowrap|ζ′(''s'' + ''it'')}} occur when {{nowrap|''s'' &gt; {{sfrac|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 {{w|''Z'' function}}, which is defined (in terms of the {{subpage|appendix|u|s=Z function and Riemann-Siegel theta function|text=Riemann-Siegel theta function}}) as:
+The absolute value of {{nowrap|ζ{{pars|{{sfrac|1|2}} + ''ig''}}|1.8}} 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 {{nowrap|ζ{{pars|{{sfrac|1|2}} + ''it''}}|1.8}} with respect to ''t'' will be zero; however this does not mean its derivative there will be zero. In fact, the {{w|Riemann hypothesis}} is equivalent to the claim that all zeros of {{nowrap|ζ′(''s'' + ''it'')}} occur when {{nowrap|''s'' &gt; {{sfrac|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 {{w|''Z'' function}}, which is defined (in terms of the {{subpage|appendix|u|s=Z function and Riemann-Siegel theta function|text=Riemann-Siegel theta function}}) as:
 <math>Z(t) = \exp(i \theta(t)) \zeta\left(\frac{1}{2} + it\right)</math>.