The Riemann zeta function and tuning: Difference between revisions

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This new function has the property that {{nowrap|Fs(''x'') {{=}} Fs(0) − Es(''x'')}}, so that all we have done is flip the sign of Es(''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. The logarithm of the {{w|Riemann zeta function}} function ~~[[The Riemann zeta function and tuning/Appendix#1a.~~ Dirichlet series for the von Mangoldt function|can be expressed]] in terms of a {{w|Dirichlet series}} involving the von Mangoldt function:

Of more interest is the fact that it is a known mathematical function. The logarithm of the {{w|Riemann zeta function}} function {{subpage|appendix|u|s=Dirichlet series for the von Mangoldt function|text=can be expressed}} in terms of a {{w|Dirichlet series}} involving the von Mangoldt function:

<math>\displaystyle

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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 ~~[[The Riemann zeta function and tuning/Appendix#2.~~ Z function and Riemann-Siegel theta function|Riemann-Siegel theta function]]) as:

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:

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

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== Removing primes ==

An [http://mathworld.wolfram.com/EulerProduct.html Euler product] formula for the Riemann zeta function ~~[[The Riemann zeta function and tuning/Appendix#1.~~ Euler product expression for the zeta function|can be easily derived]]:

An [http://mathworld.wolfram.com/EulerProduct.html Euler product] formula for the Riemann zeta function {{subpage|appendix|u|s=Euler product expression for the zeta function|text=can be easily derived}}:

<math>\displaystyle{

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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''−''s'')}} 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−''s'')ζ(''s'')}} is now dominated by 3, and the large peak values occur near equal divisions of the "tritave", ie 3.

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]]:

Along any line of constant <math>\sigma</math>, {{subpage|appendix|u|s=Conversion factor for removing primes|text=it can be shown that}}:

<math>\displaystyle{

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== Further information ==

* {{subpage|appendix|u|s=3. Black magic formulas|text=How it can be shown what ETs have a sharp vs. flat tendency}}

* {{subpage|appendix|u|s=Black magic formulas|text=How it can be shown what ETs have a sharp vs. flat tendency}}

* {{subpage|appendix|u|s=4. Computing zeta|text=How do you actually compute zeta?}}

* {{subpage|appendix|u|s=Computing zeta|text=How do you actually compute zeta?}}

== Links ==

@@ Line 107: / Line 107: @@
 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. The logarithm of the {{w|Riemann zeta function}} function [[The Riemann zeta function and tuning/Appendix#1a. Dirichlet series for the von Mangoldt function|can be expressed]] in terms of a {{w|Dirichlet series}} involving the von Mangoldt function:
+Of more interest is the fact that it is a known mathematical function. The logarithm of the {{w|Riemann zeta function}} function {{subpage|appendix|u|s=Dirichlet series for the von Mangoldt function|text=can be expressed}} in terms of a {{w|Dirichlet series}} involving the von Mangoldt function:
 <math>\displaystyle
@@ Line 329: / Line 329: @@
 === 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 [[The Riemann zeta function and tuning/Appendix#2. Z function and Riemann-Siegel theta function|Riemann-Siegel theta function]]) as:
+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:
 <math>Z(t) = \exp(i \theta(t)) \zeta\left(\frac{1}{2} + it\right)</math>.
@@ Line 610: / Line 610: @@
 == Removing primes ==
-An [http://mathworld.wolfram.com/EulerProduct.html Euler product] formula for the Riemann zeta function [[The Riemann zeta function and tuning/Appendix#1. Euler product expression for the zeta function|can be easily derived]]:
+An [http://mathworld.wolfram.com/EulerProduct.html Euler product] formula for the Riemann zeta function {{subpage|appendix|u|s=Euler product expression for the zeta function|text=can be easily derived}}:
 <math>\displaystyle{
@@ Line 618: / Line 618: @@
 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 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]]:
+Along any line of constant <math>\sigma</math>, {{subpage|appendix|u|s=Conversion factor for removing primes|text=it can be shown that}}:
 <math>\displaystyle{
@@ Line 644: / Line 644: @@
 == Further information ==
-* {{subpage|appendix|u|s=3. Black magic formulas|text=How it can be shown what ETs have a sharp vs. flat tendency}}
+* {{subpage|appendix|u|s=Black magic formulas|text=How it can be shown what ETs have a sharp vs. flat tendency}}
-* {{subpage|appendix|u|s=4. Computing zeta|text=How do you actually compute zeta?}}
+* {{subpage|appendix|u|s=Computing zeta|text=How do you actually compute zeta?}}
 == Links ==