The Riemann zeta function and tuning: Difference between revisions

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==== Gram points and zeta peaks ====

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Ludwig Siegel}} went looking for it and was able to reconstruct it after rooting industriously around in Riemann's unpublished papers.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Ludwig Siegel}} went looking for it and was able to reconstruct it after rooting industriously around in Riemann's unpublished papers.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.

=== 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}}.

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}}.

In order to define the Z function, we need first to define the {{w|Riemann–Siegel theta function}}, and in order to do that, we first need to define the [http://mathworld.wolfram.com/LogGammaFunction.html Log Gamma function]. This is not defined as the natural log of the Gamma function since that has a more complicated branch cut structure; instead, the principal branch of the Log Gamma function is defined as having a branch cut along the negative real axis, and is given by the series

@@ Line 70: / Line 70: @@
 ==== Gram points and zeta peaks ====
-Because the value of zeta increased continuously as it made its way from +&infin; 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 +&infin; 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.
 === 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}}.
+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}}.
 In order to define the Z function, we need first to define the {{w|Riemann–Siegel theta function}}, and in order to do that, we first need to define the [http://mathworld.wolfram.com/LogGammaFunction.html Log Gamma function]. This is not defined as the natural log of the Gamma function since that has a more complicated branch cut structure; instead, the principal branch of the Log Gamma function is defined as having a branch cut along the negative real axis, and is given by the series