The Riemann zeta function and tuning/Appendix
1. Euler product expression for the zeta function
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Todo: derive this |
2. Dirichlet series for the von Mangoldt function
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Todo: derive this |
3. Z function and Riemann-Siegel theta function
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 Riemann–Siegel theta function, and in order to do that, we first need to define the 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{ \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 Euler–Mascheroni constant. We now may define the Riemann–Siegel theta function as
[math]\displaystyle{ \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 z = 1 + 2it/4 into the series for Log Gamma and take the imaginary part, this yields
[math]\displaystyle{ \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 −1⁄2 and 1⁄2. It may be described for real arguments as an odd real analytic function of x, increasing when |x| > 6.29. Plots of it may be studied by use of the Wolfram online function plotter.