It is often used to compute values of the Riemann–Siegel formula, sometimes in combination with the Odlyzko–Schönhage algorithm which speeds it up considerably.
If M and N are non-negative integers, then the zeta function is equal to where is the factor appearing in the functional equation ζ(s) = γ(1 − s) ζ(1 − s), and is a contour integral whose contour starts and ends at +∞ and circles the singularities of absolute value at most 2πM.
The approximate functional equation gives an estimate for the size of the error term.
Siegel (1932)[1] and Edwards (1974) derive the Riemann–Siegel formula from this by applying the method of steepest descent to this integral to give an asymptotic expansion for the error term R(s) as a series of negative powers of Im(s).
Riemann showed that where the contour of integration is a line of slope −1 passing between 0 and 1 (Edwards 1974, 7.9).