The Riemann zeta function and tuning: Difference between revisions
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<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 \frac{z}{k} - \ln\left(1 + \frac{z}{k}\right)</math> | ||
where the lower-case gamma is [[Wikipedia: | where the lower-case gamma is the [[Wikipedia:Euler–Mascheroni constant|Euler–Mascheroni constant]]. We now may define the Riemann-Siegel theta function as | ||
<math>\theta(z) = (\Upsilon((1 + 2 i z)/4) - \Upsilon((1 - 2 i z)/4))/(2 i) - \ln(\pi) z/2</math> | <math>\theta(z) = (\Upsilon((1 + 2 i z)/4) - \Upsilon((1 - 2 i z)/4))/(2 i) - \ln(\pi) z/2</math> | ||