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...

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...

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...

Johann Peter Gustav Lejeune Dirichlet (/ˌdɪərɪˈkleɪ/; German: [ləˈʒœn diʁiˈkleː]; 13 February 1805 – 5 May 1859) was a German mathematician. In number...

(number theory) Dirichlet series (analytic number theory) Dirichlet series inversion General Dirichlet series Dirichlet space Dirichlet stability criterion...

Like the zeta function, Dirichlet series in general play an important role in analytic number theory. Generally a Dirichlet series converges if the real...

Z → C {\displaystyle \chi :\mathbb {Z} \rightarrow \mathbb {C} } is a Dirichlet character of modulus m {\displaystyle m} (where m {\displaystyle m} is...

In mathematics, Dirichlet's test is a method of testing for the convergence of a series that is especially useful for proving conditional convergence...

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...

are several integrals known as the Dirichlet integral, after the German mathematician Peter Gustav Lejeune Dirichlet, one of which is the improper integral...

Fourier series converges to the average of the left and right limits (but see Gibbs phenomenon). The Dirichlet–Dini Criterion (see Dirichlet conditions...

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:...

{\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...

In mathematics, the Dirichlet boundary condition is imposed on an ordinary or partial differential equation, such that the values that the solution takes...

Bell series, and Dirichlet series. Every sequence in principle has a generating function of each type (except that Lambert and Dirichlet series require...

the area of analytic number theory, the Dirichlet eta function is defined by the following Dirichlet series, which converges for any complex number having...

Voronoi decomposition, a Voronoi partition, or a Dirichlet tessellation (after Peter Gustav Lejeune Dirichlet). Voronoi cells are also known as Thiessen polygons...

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...

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...

In analytic number theory, a Dirichlet series, or Dirichlet generating function (DGF), of a sequence is a common way of understanding and summing arithmetic...

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...

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...

coefficients of the Dirichlet series representation of the Dedekind zeta function. The n-th coefficient of the Dirichlet series is essentially the number...

In the 14th century, the earliest examples of specific Taylor series (but not the general method) were given by Indian mathematician Madhava of Sangamagrama...

series is a series summing the terms of an infinite geometric sequence, in which the ratio of consecutive terms is constant. For example, the series 1...

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...

methods of summability, then the Dirichlet series for η defines a function on the whole complex plane — the Dirichlet eta function — and moreover, this...

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...

gamma function. If s is a complex number, the Dirichlet eta function is formed as an alternating series η ( s ) = ∑ n = 1 ∞ ( − 1 ) n − 1 n s = 1 1 s...

_{k}*\mu } The Dirichlet convolution can be defined for general arithmetic functions, and yields a ring structure, the Dirichlet ring. The Dirichlet convolution...