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


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


<math>\displaystyle\Upsilon(z) = -\gamma z - \ln z + \sum_{k=1}^\infty \left(\frac{z}{k} - \ln\left(1 + \frac{z}{k}\right)\right)</math>
The factor of <math>\exp(i \theta(t))</math> simply modifies zeta by a complex phase, and so 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 the zeros of Z in this strip correspond one to one with the zeros of ζ in the critical strip, and since θ is holomorphic on the strip with imaginary part between −{{sfrac|1|2}} and {{sfrac|1|2}}, so is Z. And Z is a real even function of the real variable ''t'', since theta was defined so as to give precisely this property.
 
where γ is the {{w|Euler–Mascheroni constant}}. We now may define the Riemann–Siegel theta function as
 
<math>\displaystyle\theta(z) = \frac{\Upsilon\left(\frac{1 + 2 i z}{4}\right) - \Upsilon\left(\frac{1 - 2 i z}{4}\right)}{2 i} - \frac{\ln(\pi)}{2} z</math>
 
Another approach is to substitute {{nowrap|''z'' {{=}} {{sfrac|1 + 2''it''|4}}}} into the series for Log Gamma and take the imaginary part, this yields
 
<math>\displaystyle \theta(t) = -\frac{\gamma + \log \pi}{2}t - \arctan 2t
+ \sum_{n=1}^\infty \left(\frac{t}{2n}
- \arctan\left(\frac{2t}{4n+1}\right)\right)</math>
 
Since the arctangent function is holomorphic in the strip with imaginary part between −1 and 1, it follows from the above formula, or arguing from the previous one, that θ is holomorphic in the strip with imaginary part between −{{frac|1|2}} and {{frac|1|2}}. It may be described for real arguments as an odd real analytic function of ''x'', increasing when {{nowrap|{{abs|''x''}} &gt; 6.29}}. Plots of it may be studied by use of the Wolfram [http://functions.wolfram.com/webMathematica/FunctionPlotting.jsp?name=RiemannSiegelTheta online function plotter].
 
Using the theta and zeta functions, we define the {{w|Z function}} as
 
<math>Z(t) = \exp(i \theta(t)) \zeta\left(\frac{1}{2} + it\right)</math>
 
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 ===
=== Plots ===
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