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In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann . One very common application is in numerical integration , i.e., approximating the area of functions or lines on a graph, where it is also known as the rectangle rule .
Riemann form; Riemann hypothesis; Riemann integral; Riemann invariant; Riemann mapping theorem; Riemann problem; Riemann series theorem; Riemann solver; Riemann sphere; Riemann sum; Riemann surface; Riemann xi function; Riemann zeta function; Riemann–Hilbert correspondence; Riemann–Hilbert problem; Riemann–Lebesgue lemma; Riemann ...
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Riemann's original use of the explicit formula was to give an exact formula for the number of primes less than a given number. To do this, take F(log(y)) to be y 1/2 /log(y) for 0 ≤ y ≤ x and 0 elsewhere. Then the main term of the sum on the right is the number of primes less than x.
The entire function ξ(s), related to the zeta function through the gamma function (or the Π function, in Riemann's usage) The discrete function J(x) defined for x ≥ 0, which is defined by J(0) = 0 and J(x) jumps by 1/n at each prime power p n. (Riemann calls this function f(x).) Among the proofs and sketches of proofs:
Riemann–Liouville integral; Riemann–Roch theorem. Arithmetic Riemann–Roch theorem; Riemann–Roch theorem for smooth manifolds; Riemann–Roch theorem for surfaces; Grothendieck–Hirzebruch–Riemann–Roch theorem; Hirzebruch–Riemann–Roch theorem; Riemann–Stieltjes integral; Riemann series theorem; Riemann sum
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.
Riemann's essay was also the starting point for Georg Cantor's work with Fourier series, which was the impetus for set theory. He also worked with hypergeometric differential equations in 1857 using complex analytical methods and presented the solutions through the behaviour of closed paths about singularities (described by the monodromy matrix ).