...inuous function of the parameters $\alpha>0$ and $\beta>0$. Used by P.G.L.
Dirichlet in his studies on the attraction of ellipsoids [[#References|[1]]]. The int
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> P.G.L.
Dirichlet, "Werke" , '''1''' , Chelsea, reprint (1969)</TD></TR><TR><TD valign="to
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''Dirichlet conditions, Dirichlet data, boundary conditions of the first kind''
are called Dirichlet boundary conditions.
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$#C+1 = 21 : ~/encyclopedia/old_files/data/D032/D.0302950 Dirichlet variational problem
The problem of finding the minimum of the [[Dirichlet integral|Dirichlet integral]]
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#REDIRECT [[Dirichlet L-function]]
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$#C+1 = 4 : ~/encyclopedia/old_files/data/D032/D.0302800 Dirichlet box principle
elements comprises at least one set with at least two elements. Dirichlet's box principle can be formulated in a most popular manner as follows: If
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A formal Dirichlet series over a ring $R$ is associated to a function $a$ from the positive in
is the [[Dirichlet convolution]] of $a$ and $b$.
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...a_n$ are real numbers and $b_n$ are complex numbers, established by P.G.L. Dirichlet and published posthumously in {{Cite|Di}}. If a sequence of real numbers $a
|valign="top"|{{Ref|Di}}|| P.G.L. Dirichlet, "Démonstration d’un théorème d’Abel", ''J. de Math. (2)'' , '''7'''
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$#C+1 = 27 : ~/encyclopedia/old_files/data/D032/D.0302930 Dirichlet series for an analytic almost\AAhperiodic function
...erent almost-periodic functions in the same strip correspond two different Dirichlet series. In the case of a $ 2 \pi $-
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...Dirichlet polynomials are partial sums of corresponding [[Dirichlet series|Dirichlet series]].
...Dirichlet $L$-function]]), as well as their powers, can be approximated by Dirichlet polynomials, mostly with $\lambda _ { m } = \operatorname { log } m$. For e
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A formal Dirichlet series over a ring $R$ is associated to a function $a$ from the positive in
is the [[Dirichlet convolution]] of $a$ and $b$.
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...convergence criterion is the [[Dirichlet criterion (convergence of series)|Dirichlet criterion (convergence of series)]].
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...ates on the hyperbola $xy = n$. The average value of $\tau(n)$ is given by Dirichlet's asymptotic formula (cf. [[Divisor problems]]).
...etic function|average value]] of the number of divisors was obtained by P. Dirichlet in 1849, in the form
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[[Dirichlet series|Dirichlet series]] or an
...finite fields, etc. The simplest representatives of $L$-functions are the Dirichlet $L$-functions (cf.
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...ed by M. Riesz [[#References|[1]]] for the summation of [[Dirichlet series|Dirichlet series]]. The method $(R,\lambda,k)$ is regular; when $\lambda_n=n$ it is e
...]</TD> <TD valign="top"> G.H. Hardy, M. Riesz, "The general theory of Dirichlet series" , Cambridge Univ. Press (1915)</TD></TR><TR><TD valign="top">[4]</
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The Dirichlet convolution of two [[arithmetic function]]s $f(n)$ and $g(n)$ is defined as
...over the positive divisors $d$ of $n$. General background material on the Dirichlet convolution can be found in, e.g., [[#References|[a1]]], [[#References|[a6]
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...n-trivial zeros of Dirichlet $ L $-functions (cf. [[Dirichlet L-function|Dirichlet $ L $-function]]), Dedekind zeta-functions (cf. [[Zeta-function|Zeta-func
In the case of Dirichlet zeta-functions the generalized Riemann hypothesis is called the extended Ri
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...)$, where $A ^ { - 1 }$ is the family of invertible elements of $A$. Hypo-Dirichlet algebras were introduced by J. Wermer [[#References|[a4]]].
...roximation of functions of a complex variable]]). Then $R ( X )$ is a hypo-Dirichlet algebra [[#References|[a3]]].
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...an be applied. Other examples are the [[Dirichlet discontinuous multiplier|Dirichlet discontinuous multiplier]], the Dirac [[Delta-function|delta-function]], et
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$#C+1 = 16 : ~/encyclopedia/old_files/data/D032/D.0302840 Dirichlet distribution
...e Dirichlet distribution: the [[Beta-distribution|beta-distribution]]. The Dirichlet distribution plays an important role in the theory of order statistics. For
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$#C+1 = 120 : ~/encyclopedia/old_files/data/D032/D.0302810 Dirichlet character
In other words, a Dirichlet character $ \mathop{\rm mod} k $
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...let problem|Dirichlet problem]]; it allows one to obtain a solution to the Dirichlet problem for a differential equation of elliptic type in domains $ D $
...other authors were dedicated to this method for finding a solution to the Dirichlet problem for the [[Laplace equation|Laplace equation]] in plane domains. The
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...a Dirichlet algebra if $A + \overline{A}$ is uniformly dense in $C ( X )$. Dirichlet algebras were introduced by A.M. Gleason [[#References|[a4]]].
...ine{\mathbf D }$. The algebra $A ( \mathbf{D} )$ is a typical example of a Dirichlet algebra on the unit circle $\partial \mathbf{D}$. For $A ( \mathbf{D} )$, t
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''Dirichlet–Laplace operator''
...$C _ { 0 } ^ { \infty } ( \Omega )$) is given by the [[Dirichlet integral|Dirichlet integral]]
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The [[Dirichlet convolution]] product
Formally, the [[Dirichlet series]] of a multiplicative function $f$ has an [[Euler product]]:
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is infinite and has [[Dirichlet density|Dirichlet density]] $\# A / n$, where $n = [ L : K ]$.
...specifies in addition that $P _ { A }$ is regular (see [[Dirichlet density|Dirichlet density]]) and that
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The homogeneous Dirichlet problem in a disc $ C $:
...infinite number of linearly independent solutions [[#References|[1]]]. The Dirichlet problem for the inhomogeneous equation $ w _ {\overline{z}\; \overline{z}
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...Suppose, for example, that it is required to solve the [[Dirichlet problem|Dirichlet problem]] for the [[Poisson equation|Poisson equation]] $ \Delta u = - 2
For the solution of the Dirichlet problem for the Poisson equation $ \Delta u = - 4 \pi \rho ( x, y, z) $
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...haviour of a real sequence to the analytic properties of the associated [[Dirichlet series]]. It is used in the study of [[arithmetic function]]s and yields
An important number-theoretic application of the theorem is to [[Dirichlet series]] of the form $\sum_{n=1}^\infty a(n) n^{-s}$ where $a(n)$ is non-n
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