The Riemann zeta function and tuning/Appendix: Difference between revisions
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== | == 3. Z function and Riemann-Siegel theta function == | ||
Below proceeds a | Below proceeds a mathematically rigorous exposition of the Z function and theta function, cut from Gene Ward Smith's derivation for the sake of clarifying the actual steps taken. | ||
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 | 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 | ||