Incidentally, this relation is interesting also because it actually exhibits ζ ( s ) as a Dirichlet series ( of the η-function ) which is convergent ( albeit non-absolutely ) in the larger half-plane σ > 0 ( not just σ > 1 ), up to an elementary factor.

According to the definition above, some exotic functions, for example the Dirichlet function, are also periodic ; in the case of Dirichlet function, any nonzero rational number is a period.

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 infinitely many primes of the form a + nd, where n ≥ 0.

It is named after Georgy Voronoi, and is also called a Voronoi tessellation, a Voronoi decomposition, or a Dirichlet tessellation ( after Lejeune Dirichlet ).

The Dirichlet convolution of two multiplicative functions is again multiplicative, and every multiplicative function has a Dirichlet inverse that is also multiplicative.

Dirichlet also proved that for prime q ≡ 3 ( mod 4 ),

Some sort of reversible-jump variant is also needed when doing MCMC or Gibbs sampling over nonparametric Bayesian models such as those involving the Dirichlet process or Chinese restaurant process, where the number of mixing components / clusters / etc.

His lecture at the Academy has also put Dirichlet in close contact with Fourier and Poisson, who raised his interest in theoretical physics, especially Fourier's analytic theory of heat.

Humboldt also secured a recommendation letter from Gauss, who upon reading his memoir on Fermat's theorem wrote with an unusual amount of praise that " Dirichlet showed excellent talent ".

Dirichlet enjoyed his time in Göttingen as the lighter teaching load allowed him more time for research and, also, he got in close contact with the new generation of researchers, especially Richard Dedekind and Bernhard Riemann.

It also introduced the Dirichlet function as an example that not any function is integrable ( the definite integral was still a developing topic at the time ) and, in the proof of the theorem for the Fourier series, introduced the Dirichlet kernel and the Dirichlet integral.

Riemann later named this approach the Dirichlet principle, although he knew it had also been used by Gauss and by Lord Kelvin.

Dirichlet also worked in mathematical physics, lecturing and publishing research in potential theory ( including the Dirichlet problem and Dirichlet principle mentioned above ), the theory of heat and hydrodynamics.

Gradient estimates were also used crucially in Yau's joint work with S. Y. Cheng to give a complete proof of the higher dimensional Hermann Minkowski problem and the Dirichlet problem for the real Monge – Ampère equation, and other results on the Kähler – Einstein metric of bounded pseudoconvex domains.

One example of such a function is the indicator function of the rational numbers, also known as the Dirichlet function, named after German mathematician Peter Gustav Lejeune Dirichlet.

Clausius graduated from the University of Berlin in 1844 where he studied mathematics and physics with, among others, Gustav Magnus, Johann Dirichlet and Jakob Steiner.

Back in Berlin, Kronecker studied mathematics with Gustav Lejeune Dirichlet and in 1845 defended his dissertation in algebraic number theory written under Dirichlet's supervision.

Among the earliest boundary value problems to be studied is the Dirichlet problem, of finding the harmonic functions ( solutions to Laplace's equation ); the solution was given by the Dirichlet's principle.

He studied at the University of Berlin under Dirichlet in 1836 and at the University of Königsberg in 1839.

String compactifications studied by Harvey and Minahan, Ishibashi and Onogi, and Pradisi and Sagnotti in 1987-89 also employed Dirichlet boundary conditions.

Heine provided the first published definition of uniform continuity in 1872, but based these ideas on lectures given by Dirichlet in 1854.

In the language of Dirichlet convolutions, the first formula may be written as

1912 Plemelj published a very simple proof for the Fermat's last theorem for exponent n = 5, which was first given almost simultaneously by Dirichlet in 1828 and Legendre in 1830.

Dirichlet was the first to prove this and implicitly he used the existence of a finite subcover of a given open cover of a closed interval in his proof.

The generalized Riemann hypothesis ( for Dirichlet L-functions ) was probably formulated for the first time by Piltz in 1884.

Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859 ) was a German mathematician with 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.

Dirichlet had a good reputation with students for the clarity of his explanations and enjoyed teaching, especially as his University lectures tended to be on the more advanced topics in which he was doing research: number theory ( he was the first German professor to give lectures on number theory ), analysis and mathematical physics.

The Dirichlet divisor problem, for which he found the first results, is still an unsolved problem in number theory despite later contributions by other researchers.

It is often said to have begun with Dirichlet's introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic progressions.

More generally, the same question can be asked about the number of primes in any arithmetic progression a + nq for any integer n. In one of the first applications of analytic techniques to number theory, Dirichlet proved that any arithmetic progression with a and q coprime contains infinitely many primes.

According to a theorem on uniform convergence of Dirichlet series first proven by Cahen in 1894, the function is then analytic for, a region which includes the line

It is widely believed that Kummer was led to his " ideal complex numbers " by his interest in Fermat's Last Theorem ; there is even a story often told that Kummer, like Lamé, believed he had proven Fermat's Last Theorem until Dirichlet told him his argument relied on unique factorization ; but the story was first told by Kurt Hensel in 1910 and the evidence indicates it likely derives from a confusion by one of Hensel's sources.

Mixed Dirichlet / Neumann boundary conditions were first considered by Warren Siegel in 1976 as a means of lowering the critical dimension of open string theory from 26 or 10 to 4 ( Siegel also cites unpublished work by Halpern, and a 1974 paper by Chodos and Thorn, but a reading of the latter paper shows that it is actually concerned with linear dilation backgrounds, not Dirichlet boundary conditions ).

Douglas is best known for the development of matrix models ( the first nonperturbative formulations of string theory ), for his work on Dirichlet branes and on noncommutative geometry in string theory, and for the development of the statistical approach to string phenomenology.

Although other mathematicians had conjectured similar formulae, Dirichlet gave the first rigorous proof.

The contains two key results in number theory which were first proved by Dirichlet.

This fact can be used to prove existence and uniqueness of solutions to the Dirichlet problem for the Poisson equation in suitably regular domains, and for suitably well-behaved functions ƒ: one first applies a Newtonian potential to obtain a solution, and then adjusts by adding a harmonic function to get the correct boundary data.

Let K be an algebraic number field K. Its Dedekind zeta function is first defined for complex numbers s with real part Re ( s ) > 1 by the Dirichlet series

The first paper where a set of ( fairly complicate ) necessary and sufficient conditions for the solvability of the Dirichlet problem for holomorphic functions of several variables is given.

The Dirichlet problem for Laplace's equation consists of finding a solution on some domain such that on the boundary of is equal to some given function.

A good example of this is one-dimensional heat transport with Dirichlet boundary conditions, the solution of which can be written as a time-dependent linear combination of sinusoids of differing frequencies ; this makes solutions very flexible.

Solutions of boundary value problems for the Laplace equation satisfy the Dirichlet principle.

: Energy definition: A conformal immersion X: M → R < sup > 3 </ sup > is minimal if and only if it is a critical point of the Dirichlet energy for all compactly supported variations, or equivalently if any point p ∈ M has a neighbourhood with least energy relative to its boundary.

Using this expression, it is possible to solve Laplace's equation or Poisson's equation, subject to either Neumann or Dirichlet boundary conditions.

In other words, we can solve for everywhere inside a volume where either ( 1 ) the value of is specified on the bounding surface of the volume ( Dirichlet boundary conditions ), or ( 2 ) the normal derivative of is specified on the bounding surface ( Neumann boundary conditions ).

If the problem is to solve a Dirichlet boundary value problem, the Green's function should be chosen such that vanishes when either x or x < nowiki >'</ nowiki > is on the bounding surface. Thus only one of the two terms in the surface integral remains.

In mathematics, the Dirichlet ( or first-type ) boundary condition is a type of boundary condition, named after Johann Peter Gustav Lejeune Dirichlet ( 1805 – 1859 ).

the Dirichlet boundary conditions on the interval take the form:

where denotes the Laplacian, the Dirichlet boundary conditions on a domain take the form:

For example, there is the Cauchy boundary condition, or the mixed boundary condition which is a combination of the Dirichlet and Neumann conditions.

If the boundary gives a value to the problem then it is a Dirichlet boundary condition.

* Dirichlet boundary condition

In string theory, D-branes are a class of extended objects upon which open strings can end with Dirichlet boundary conditions, after which they are named.

The equations of motion of string theory require that the endpoints of an open string ( a string with endpoints ) satisfy one of two types of boundary conditions: The Neumann boundary condition, corresponding to free endpoints moving through spacetime at the speed of light, or the Dirichlet boundary conditions, which pin the string endpoint.