Z Function - The Riemann-Siegel Formula

The Riemann-Siegel Formula

Calculation of the value of Z(t) for real t, and hence of the zeta-function along the critical line, is greatly expedited by the Riemann-Siegel formula. This formula tells us

where the error term R(t) has a complex asymptotic expression in terms of the function

and its derivatives. If, and then

$R(t) \sim (-1)^{N-1} \left( \Psi(p)u^{-1} - \frac{1}{96 \pi^2}\Psi^{(3)}(p)u^{-3} + \cdots\right)$

where the ellipsis indicates we may continue on to higher and increasingly complex terms.

Other efficient series for Z(t) are known, in particular several using the incomplete gamma function. If

then an especially nice example is

$Z(t) =2 \Re \left(e^{i \theta(t)} \left(\sum_{n=1}^\infty Q\left(\frac{s}{2},\pi i n^2 \right) - \frac{\pi^{s/2} e^{\pi i s/4}} {s \Gamma\left(\frac{s}{2}\right)} \right)\right)$