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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
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.
Cartier (1982) discussed a related example, where due to a bizarre bug a computer program listed zeros of the Riemann zeta function as eigenvalues of the same Laplacian operator. Schumayer & Hutchinson (2011) surveyed some of the attempts to construct a suitable physical model related to the Riemann zeta function.
Riemann function may refer to one of the several functions named after the mathematician Bernhard Riemann, including: Riemann zeta function; Thomae's function, also called the Riemann function; Riemann theta function, Riemann's R, an approximation of the prime-counting function π(x), see Prime-counting function#Exact form. Almost nowhere ...
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.
Riemann used asymptotics of the theta function to obtain the analytic continuation, and the automorphy of the theta function to prove the functional equation. Erich Hecke, and later Hans Maass, applied the same Mellin transform method to modular forms on the upper half-plane, after which Riemann's example can be seen as a special case.
The solutions are denoted by the Riemann P-symbol (also known as the Papperitz symbol) = {′ ′ ′}The standard hypergeometric function may be expressed as (,;;) = {}The P-functions obey a number of identities; one of them allows a general P-function to be expressed in terms of the hypergeometric function.