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)

Read more about this topic:  Z Function

Famous quotes containing the word formula:

    “Ideals possess the strange quality that if they were completely realized they would turn into nonsense. One could easily follow a commandment such as “Thou shalt not kill” to the point of dying of starvation; and I might establish the formula that for the proper functioning of the mesh of our ideals, as in the case of a strainer, the holes are just as important as the mesh.”
    —Robert Musil (1880–1942)