The Riemann zeta function and tuning: Difference between revisions

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

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

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== Further information ==

* [[The Riemann zeta function and tuning/Appendix#5. Black magic formulas|How it can be shown what ETs have a sharp vs. flat tendency]]

* [[The Riemann zeta function and tuning/Appendix#3. Black magic formulas|How it can be shown what ETs have a sharp vs. flat tendency]]

* [[The Riemann zeta function and tuning/Appendix#6. Computing zeta|How do you actually compute zeta?]]

* [[The Riemann zeta function and tuning/Appendix#4. Computing zeta|How do you actually compute zeta?]]

== Links ==

@@ Line 331: / Line 331: @@
 === 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#3. 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 [[The Riemann zeta function and tuning/Appendix#2. Z function and Riemann-Siegel theta function|Riemann-Siegel theta function]]) as:
 <math>Z(t) = \exp(i \theta(t)) \zeta\left(\frac{1}{2} + it\right)</math>.
@@ Line 706: / Line 706: @@
 == Further information ==
-* [[The Riemann zeta function and tuning/Appendix#5. Black magic formulas|How it can be shown what ETs have a sharp vs. flat tendency]]
+* [[The Riemann zeta function and tuning/Appendix#3. Black magic formulas|How it can be shown what ETs have a sharp vs. flat tendency]]
-* [[The Riemann zeta function and tuning/Appendix#6. Computing zeta|How do you actually compute zeta?]]
+* [[The Riemann zeta function and tuning/Appendix#4. Computing zeta|How do you actually compute zeta?]]
 == Links ==