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
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
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“So, if we must give a general formula applicable to all kinds of soul, we must describe it as the first actuality [entelechy] of a natural organized body.”
—Aristotle (384323 B.C.)