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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 Z function is a function used for studying the Riemann zeta function along the critical line where the argument is one-half. It is also called the Riemann–Siegel Z function, the Riemann–Siegel zeta function, the Hardy function, the Hardy Z function and the Hardy zeta function.
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.
In mathematics, the method of steepest descent or saddle-point method is an extension of Laplace's method for approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point (saddle point), in roughly the direction of steepest descent or stationary phase.
Generally speaking, Riemann solvers are specific methods for computing the numerical flux across a discontinuity in the Riemann problem. [1] They form an important part of high-resolution schemes; typically the right and left states for the Riemann problem are calculated using some form of nonlinear reconstruction, such as a flux limiter or a WENO method, and then used as the input for the ...
This book concerns the Riemann zeta function and the Riemann hypothesis on the location of the zeros of this function. It includes a translation of Riemann's original paper on these subjects, and analyzes this paper in depth; it also covers methods of computing the function such as Euler–Maclaurin summation and the Riemann–Siegel formula.
In mathematics, the local zeta function Z(V, s) (sometimes called the congruent zeta function or the Hasse–Weil zeta function) is defined as (,) = (= ())where V is a non-singular n-dimensional projective algebraic variety over the field F q with q elements and N k is the number of points of V defined over the finite field extension F q k of F q.
The function has the series expansion = = +, where = ()! [ ()] | = = [()], where the sum extends over ρ, the non-trivial zeros of the zeta function, in order of | |.. This expansion plays a particularly important role in Li's criterion, which states that the Riemann hypothesis is equivalent to having λ n > 0 for all positive n.