In the field of mathematical analysis, a general Dirichlet series is an infinite series that takes the form of ∑ n = 1 ∞ a n e − λ n s , {\displaystyle...
10 KB (1,999 words) - 18:39, 27 September 2023
a_{n}} is a complex sequence. It is a special case of general Dirichlet series. Dirichlet series play a variety of important roles in analytic number theory...
25 KB (5,275 words) - 14:39, 5 March 2024
(number theory) Dirichlet series (analytic number theory) Dirichlet series inversion General Dirichlet series Dirichlet space Dirichlet stability criterion...
3 KB (224 words) - 16:08, 20 March 2022
Like the zeta function, Dirichlet series in general play an important role in analytic number theory. Generally a Dirichlet series converges if the real...
58 KB (9,694 words) - 15:11, 3 September 2024
series expansion on an annulus centered at the singularity. A general Dirichlet series is a series of the form ∑ n = 1 ∞ a n e − λ n s . {\textstyle \sum _{n=1}^{\infty...
9 KB (968 words) - 12:14, 9 July 2024
Johann Peter Gustav Lejeune Dirichlet (/ˌdɪərɪˈkleɪ/; German: [ləˈʒœn diʁiˈkleː]; 13 February 1805 – 5 May 1859) was a German mathematician. In number...
31 KB (3,581 words) - 16:53, 3 September 2024
mathematics, Dirichlet's test is a method of testing for the convergence of a series. It is named after its author Peter Gustav Lejeune Dirichlet, and was...
4 KB (903 words) - 11:19, 6 September 2024
are several integrals known as the Dirichlet integral, after the German mathematician Peter Gustav Lejeune Dirichlet, one of which is the improper integral...
14 KB (2,904 words) - 07:31, 3 February 2024
In mathematics, the Dirichlet boundary condition is imposed on an ordinary or partial differential equation, such that the values that the solution takes...
4 KB (435 words) - 14:50, 29 May 2024
Leibniz formula for π (redirect from Gregory-Leibniz series)
{\textstyle \arctan 1={\tfrac {1}{4}}\pi .} It also is the Dirichlet L-series of the non-principal Dirichlet character of modulus 4 evaluated at s = 1 , {\displaystyle...
9 KB (1,509 words) - 08:03, 4 September 2024
Z → C {\displaystyle \chi :\mathbb {Z} \rightarrow \mathbb {C} } is a Dirichlet character of modulus m {\displaystyle m} (where m {\displaystyle m} is...
48 KB (11,535 words) - 03:28, 8 August 2024
Fourier series converges to the average of the left and right limits (but see Gibbs phenomenon). The Dirichlet–Dini Criterion (see Dirichlet conditions...
22 KB (4,043 words) - 09:31, 28 August 2024
In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers a and d, there...
22 KB (2,881 words) - 06:49, 8 September 2024
Pigeonhole principle (redirect from Dirichlet's Box Principle)
commonly called Dirichlet's box principle or Dirichlet's drawer principle after an 1834 treatment of the principle by Peter Gustav Lejeune Dirichlet under the...
31 KB (4,141 words) - 12:35, 26 August 2024
Generating function (redirect from Dirichlet generating function)
Bell series, and Dirichlet series. Every sequence in principle has a generating function of each type (except that Lambert and Dirichlet series require...
87 KB (14,332 words) - 03:20, 1 September 2024
Voronoi diagram (redirect from Dirichlet domain)
Voronoi decomposition, a Voronoi partition, or a Dirichlet tessellation (after Peter Gustav Lejeune Dirichlet). Voronoi cells are also known as Thiessen polygons...
46 KB (5,593 words) - 23:43, 11 August 2024
In analytic number theory, a Dirichlet series, or Dirichlet generating function (DGF), of a sequence is a common way of understanding and summing arithmetic...
16 KB (3,812 words) - 23:29, 26 March 2023
naturally in the study of Dirichlet series such as the Riemann zeta function. It describes the multiplication of two Dirichlet series in terms of their coefficients:...
16 KB (2,548 words) - 22:31, 29 July 2024
the area of analytic number theory, the Dirichlet eta function is defined by the following Dirichlet series, which converges for any complex number having...
19 KB (3,707 words) - 16:11, 30 June 2024
L-function (section Rise of the general theory)
generalisations of the Riemann zeta function and the L-series for a Dirichlet character are constructed, and their general properties, in most cases still out of reach...
8 KB (984 words) - 11:59, 7 May 2024
instance of the Dirichlet theorem for Fourier series. This example leads to a solution of the Basel problem. A proof that a Fourier series is a valid representation...
70 KB (10,863 words) - 12:21, 11 September 2024
determined by a combination of inflation rates and interest rates. In general, a geometric series is written as a + a r + a r 2 + a r 3 + . . . {\displaystyle...
65 KB (9,899 words) - 19:21, 9 September 2024
In the 14th century, the earliest examples of specific Taylor series (but not the general method) were given by Indian mathematician Madhava of Sangamagrama...
48 KB (8,253 words) - 02:02, 23 August 2024
Class number formula (redirect from Dirichlet_class_number_formula)
coefficients of the Dirichlet series representation of the Dedekind zeta function. The n-th coefficient of the Dirichlet series is essentially the number...
9 KB (1,302 words) - 16:28, 2 July 2024
They occur, for example, in the functional equations of Dirichlet L-functions, where for a Dirichlet character χ the equation relating L(s, χ) and L(1 − s...
7 KB (918 words) - 18:21, 8 June 2023
methods of summability, then the Dirichlet series for η defines a function on the whole complex plane — the Dirichlet eta function — and moreover, this...
12 KB (1,719 words) - 08:53, 7 November 2023
Analytic number theory (section Dirichlet series)
begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic progressions...
27 KB (3,825 words) - 07:06, 21 July 2024
Divisor summatory function (redirect from Dirichlet divisor problem)
can be proven using the Dirichlet hyperbola method, and was first established by Dirichlet in 1849.: 37–38, 69 The Dirichlet divisor problem, precisely...
11 KB (1,936 words) - 16:54, 8 April 2024
some convergent alternating series may fail the first part of the test. For a generalization, see Dirichlet's test. A series of the form ∑ n = 0 ∞ ( − 1...
8 KB (1,453 words) - 19:51, 24 June 2024
placing hierarchical Dirichlet process priors over the HMM transition matrix. Step detection Keogh, Eamonn, et al. "Segmenting time series: A survey and novel...
5 KB (665 words) - 20:52, 12 June 2024