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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.
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.
In mathematical physics, in particular electromagnetism, the Riemann–Silberstein vector [1] or Weber vector [2] [3] named after Bernhard Riemann, Heinrich Martin Weber and Ludwik Silberstein, (or sometimes ambiguously called the "electromagnetic field") is a complex vector that combines the electric field E and the magnetic field B.
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.
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.
B. Riemann used the Riemann–Siegel formula (unpublished, but reported in Siegel 1932). 1903 15 J. P. Gram (1903) used Euler–Maclaurin formula and discovered Gram's law. He showed that all 10 zeros with imaginary part at most 50 range lie on the critical line with real part 1/2 by computing the sum of the inverse 10th powers of the roots he ...
The Weyl tensor has the same basic symmetries as the Riemann tensor, but its 'analogue' of the Ricci tensor is zero: = = = = The Ricci tensor, the Einstein tensor, and the traceless Ricci tensor are symmetric 2-tensors: