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Siegel derived it from the Riemann–Siegel integral formula, an expression for the zeta function involving contour integrals. 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.
In mathematics, the Riemann–Siegel theta function is defined in terms of the gamma function as = ((+)) for real values of t.Here the argument is chosen in such a way that a continuous function is obtained and () = holds, i.e., in the same way that the principal branch of the log-gamma function is defined.
It is an even function, and real analytic for real values. It follows from the fact that the Riemann–Siegel theta function and the Riemann zeta function are both holomorphic in the critical strip, where the imaginary part of t is between −1/2 and 1/2, that the
where Hardy's Z function and the Riemann–Siegel theta function θ are uniquely defined by this and the condition that they are smooth real functions with θ(0) = 0. By finding many intervals where the function Z changes sign one can show that there are many zeros on the critical line.
The Riemann–Siegel formula used for calculating the Riemann zeta function with imaginary part T uses a finite Dirichlet series with about N = T 1/2 terms, so when finding about N values of the Riemann zeta function it is sped up by a factor of about T 1/2.
Zeta functions and L-functions express important relations between the geometry of Riemann surfaces, number theory and dynamical systems.Zeta functions, and their generalizations such as the Selberg class S, are conjectured to have various important properties, including generalizations of the Riemann hypothesis and various relationships with automorphic forms as well as to the representations ...
The analytic continuation of this zeta function ζ to all complex s ≠ 1; The entire function ξ(s), related to the zeta function through the gamma function (or the Π function, in Riemann's usage) The discrete function J(x) defined for x ≥ 0, which is defined by J(0) = 0 and J(x) jumps by 1/n at each prime power p n. (Riemann calls this ...
Riemann–Siegel formula; Riemann–Siegel theta function; Riemann–Silberstein vector; Riemann–Stieltjes integral; Riemann–von Mangoldt formula;