The Riemann zeta function and tuning: Difference between revisions
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<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> | <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'' {{=}} {{ | 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 | <math>\displaystyle \theta(t) = -\frac{\gamma + \log \pi}{2}t - \arctan 2t | ||