Search results
Results from the WOW.Com Content Network
In mathematics, the Dirichlet function [1] [2] is the indicator function of the set of rational numbers, i.e. () = if x is a rational number and () = if x is not a rational number (i.e. is an irrational number).
The Dirichlet L-function L(s, χ) = 1 − 3 −s + 5 −s − 7 −s + ⋅⋅⋅ (sometimes given the special name Dirichlet beta function), with trivial zeros at the negative odd integers. Let χ be a primitive character modulo q, with q > 1. There are no zeros of L(s, χ) with Re(s) > 1. For Re(s) < 0, there are zeros at certain negative ...
One example of such a function is the indicator function of the rational numbers, also known as the Dirichlet function. This function is denoted as 1 Q {\displaystyle \mathbf {1} _{\mathbb {Q} }} and has domain and codomain both equal to the real numbers .
4.1.1 Examples m = 3, 5, 7, 9. 4.2 Powers of 2. ... It is not necessary to establish the defining properties 1) – 3) to show that a function is a Dirichlet character.
In mathematics, Dirichlet's test is a method of testing for the convergence of a series that is especially useful for proving conditional convergence. It is named after its author Peter Gustav Lejeune Dirichlet , and was published posthumously in the Journal de Mathématiques Pures et Appliquées in 1862.
Let σ 0 (n) be the divisor-counting function, and let D(n) be its summatory function: = = (). Computing D(n) naïvely requires factoring every integer in the interval [1, n]; an improvement can be made by using a modified Sieve of Eratosthenes, but this still requires Õ(n) time.
By the transfer principle, the natural extension of the Dirichlet function takes the value 1 at a n. Note that the hyperrational point a n is infinitely close to π. Thus the natural extension of the Dirichlet function takes different values (0 and 1) at these two infinitely close points, and therefore the Dirichlet function is not continuous ...
The set of arithmetic functions forms a commutative ring, the Dirichlet ring, under pointwise addition, where f + g is defined by (f + g)(n) = f(n) + g(n), and Dirichlet convolution. The multiplicative identity is the unit function ε defined by ε ( n ) = 1 if n = 1 and ε ( n ) = 0 if n > 1 .