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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.
One popular restriction is the use of "left-hand" and "right-hand" Riemann sums. In a left-hand Riemann sum, t i = x i for all i, and in a right-hand Riemann sum, t i = x i + 1 for all i. Alone this restriction does not impose a problem: we can refine any partition in a way that makes it a left-hand or right-hand sum by subdividing it at each t i.
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.
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 Itô integral can be defined in a manner similar to the Riemann–Stieltjes integral, that is as a limit in probability of Riemann sums; such a limit does not necessarily exist pathwise. Suppose that B is a Wiener process (Brownian motion) and that H is a right-continuous ( càdlàg ), adapted and locally bounded process.
The trapezoidal rule may be viewed as the result obtained by averaging the left and right Riemann sums, and is sometimes defined this way. The integral can be even better approximated by partitioning the integration interval, applying the trapezoidal rule to each subinterval, and summing the results. In practice, this "chained" (or "composite ...
Suppose the cross-section is defined by the graph of the positive function f(x) on the interval [a, b]. Then the formula for the volume will be: If the function is of the y coordinate and the axis of rotation is the x-axis then the formula becomes:
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.