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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.
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.
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
The Riemann zeta function ζ(z) plotted with domain coloring. [1] The pole at = and two zeros on the critical line.. The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter ζ (), is a mathematical function of a complex variable defined as () = = = + + + for >, and its analytic continuation elsewhere.
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.
A prototypical example, the Riemann zeta function has a functional equation relating its value at the complex number s with its value at 1 − s. In every case this relates to some value ζ(s) that is only defined by analytic continuation from the infinite series definition.
In mathematics, more specifically in the field of analytic number theory, a Landau–Siegel zero or simply Siegel zero, also known as exceptional zero [1]), named after Edmund Landau and Carl Ludwig Siegel, is a type of potential counterexample to the generalized Riemann hypothesis, on the zeros of Dirichlet L-functions associated to quadratic number fields.