The Riemann zeta function and tuning: Difference between revisions

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In order to define the Z function, we need first to define the [[Wikipedia:Riemann-Siegel theta function|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>\displaystyle\Upsilon(z) = -\gamma z - \ln z + \sum_{k=1}^\infty \frac{z}{k} - \ln\left(1 + \frac{z}{k}\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>

where γ is the [[Wikipedia:Euler–Mascheroni constant|Euler–Mascheroni constant]]. We now may define the Riemann-Siegel theta function as

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 In order to define the Z function, we need first to define the [[Wikipedia:Riemann-Siegel theta function|Riemann&ndash;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>\displaystyle\Upsilon(z) = -\gamma z - \ln z + \sum_{k=1}^\infty \frac{z}{k} - \ln\left(1 + \frac{z}{k}\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>
 where &gamma; is the [[Wikipedia:Euler–Mascheroni constant|Euler&ndash;Mascheroni constant]]. We now may define the Riemann-Siegel theta function as