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Dirichlet Hyperbola Method
In number theory, the Dirichlet hyperbola method is a technique to evaluate the sum : F(n) = \sum_^n f(k), where f is a multiplicative function. The first step is to find a pair of multiplicative functions g and h such that, using Dirichlet convolution, we have f = g \ast h; the sum then becomes : F(n) = \sum_^n \sum_^ g(x) h(y), where the inner sum runs over all ordered pairs (x,y) of positive integers such that xy=k. In the Cartesian plane, these pairs lie on a hyperbola, and when the double sum is fully expanded, there is a bijection between the terms of the sum and the lattice points in the first quadrant on the hyperbolas of the form xy=k, where k runs over the integers 1 \leq k \leq n: for each such point (x,y), the sum contains a term g(x)h(y), and vice versa. Let a be a real number, not necessarily an integer, such that 1 < a < n, and let . Then the lattice points can be split into three overlapping regions: one region is bounded by |
Dirichlet Hyperbola Example 4
Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and other topics in mathematical analysis; he is credited with being one of the first mathematicians to give the modern formal definition of a function. Although his surname is Lejeune Dirichlet, he is commonly referred to by his mononym Dirichlet, in particular for results named after him. Biography Early life (1805–1822) Gustav Lejeune Dirichlet was born on 13 February 1805 in Düren, a town on the left bank of the Rhine which at the time was part of the First French Empire, reverting to Prussia after the Congress of Vienna in 1815. His father Johann Arnold Lejeune Dirichlet was the postmaster, merchant, and city councilor. His paternal grandfather had come to Düren from Richelette (or more likely Richelle), a small community north eas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
