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While not derived as a Riemann sum, taking the average of the left and right Riemann sums is the trapezoidal rule and gives a trapezoidal sum. It is one of the simplest of a very general way of approximating integrals using weighted averages. This is followed in complexity by Simpson's rule and Newton–Cotes formulas.
In mathematics, a zeta function is (usually) a function analogous to the original example, the Riemann zeta function = =. Zeta functions include: Airy zeta function, related to the zeros of the Airy function; Arakawa–Kaneko zeta function; Arithmetic zeta function
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:
For example, the full zeta function exists at = (and is therefore finite there), but the corresponding series would be + + + …, whose partial sums would grow indefinitely large. The zeta function values listed below include function values at the negative even numbers ( s = −2 , −4 , etc. ), for which ζ ( s ) = 0 and which make up the so ...
An example of Riemann sums for the integral ... Riemann sum; Global file usage. The following other wikis use this file: Usage on de.wikibooks.org
A partition of an interval being used in a Riemann sum. The partition itself is shown in grey at the bottom, with the norm of the partition indicated in red. In mathematics, a partition of an interval [a, b] on the real line is a finite sequence x 0, x 1, x 2, …, x n of real numbers such that a = x 0 < x 1 < x 2 < … < x n = b.
The Riemann zeta function can be replaced by a Dirichlet L-function of a Dirichlet character χ. The sum over prime powers then gets extra factors of χ(p m), and the terms Φ(1) and Φ(0) disappear because the L-series has no poles.
In number theory, the divisor summatory function is a function that is a sum over the divisor function. It frequently occurs in the study of the asymptotic behaviour of the Riemann zeta function. The various studies of the behaviour of the divisor function are sometimes called divisor problems.