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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
:<math>Z(t) = 2 \sum_{n^2 < t/2\pi} n^{-1/2}\cos(\theta(t)-t \log n) +R(t),</math>
where the error term R(t) has a complex asymptotic expression in terms of the function
:<math>\Psi(z) = \frac{cos 2\pi(z^2-z-1/16)}{cos 2\pi z}</math>
and its derivatives. If <math>u=(\frac{t}{2\pi})^{1/4}</math>,<math>N=\lfloor u^2 \rfloor</math> and <math>p = u^2 - N</math> then
:<math>R(t) \sim (-1)^{N-1}
\left( \Psi(p)u^{-1}
- \frac{1}{96 \pi^2}\Psi^{(3)}(p)u^{-2}
+ \cdots\right)</math>
where the ellipsis indicates we may continue on to higher and increasingly complex terms.
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