Heine proposed that Cantor solve an open problem that had eluded Dirichlet, Lipschitz, Bernhard Riemann, and Heine himself: the uniqueness of the representation of a function by trigonometric series.

More examples are shown in the article on Dirichlet series.

The generated sequences can perhaps be more easily understood by considering the corresponding Dirichlet series: each repeated application of the transform corresponds to multiplication by the Riemann zeta function.

In 1838 Johann Peter Gustav Lejeune Dirichlet came up with his own approximating function, the logarithmic integral li ( x ) ( under the slightly different form of a series, which he communicated to Gauss ).

In a handwritten note on a reprint of his 1838 paper " Sur l ' usage des séries infinies dans la théorie des nombres ", which he mailed to Carl Friedrich Gauss, Johann Peter Gustav Lejeune Dirichlet conjectured ( under a slightly different form appealing to a series rather than an integral ) that an even better approximation to π ( x ) is given by the offset logarithmic integral function Li ( x ), defined by

Many generalizations of the Riemann zeta function, such as Dirichlet series, Dirichlet L-functions and L-functions, are known.

The above series is a prototypical Dirichlet series that converges absolutely to an analytic function for s such that and diverges for all other values of s. Riemann showed that the function defined by the series on the half-plane of convergence can be continued analytically to all complex values.

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.

The Dirichlet series that generates the square-free numbers is

He expounds on Fourier series, Cantor-Riemann theory, the Poisson integral and the Dirichlet problem.

The Dirichlet series for the Liouville function gives the Riemann zeta function as

There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series ; definitions and examples are given below.

Every sequence in principle has a generating function of each type ( except that Lambert and Dirichlet series require indices to start at 1 rather than 0 ), but the ease with which they can be handled may differ considerably.

Formal Dirichlet series are often classified as generating functions, although they are not strictly formal power series.

The Dirichlet series generating function of a sequence a < sub > n </ sub > is

The Dirichlet series generating function is especially useful when a < sub > n </ sub > is a multiplicative function, when it has an Euler product expression in terms of the function's Bell series

If a < sub > n </ sub > is a Dirichlet character then its Dirichlet series generating function is called a Dirichlet L-series.

where * denotes the Dirichlet convolution, and 1 is the constant function.

The theorem follows because * is ( commutative and ) associative, and 1 * μ = i, where i is the identity function for the Dirichlet convolution, taking values i ( 1 )= 1, i ( n )= 0 for all n > 1.

* Neutral vector ( statistics ), a multivariate random variable is neutral if it exhibits a particular type of statistical independence seen when considering the Dirichlet distribution

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.

where β is the Dirichlet beta function.

However, there are other applications where there is a need to describe the uncertainty with which a function is known and where the state of knowledge about the true function can be expressed by saying that it is an unknown realisation of a random function, for example in the Dirichlet process.

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.

The Jacobi symbol is a Dirichlet character to the modulus n.

The Liouville function's Dirichlet inverse is the absolute value of the Mobius 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.

When the Riemann hypothesis is formulated for Dedekind zeta-functions, it is known as the extended Riemann hypothesis ( ERH ) and when it is formulated for Dirichlet L-functions, it is known as the generalized Riemann hypothesis ( GRH ).

A Dirichlet character is a completely multiplicative arithmetic function χ such that there exists a positive integer k with χ ( n + k )

If such a character is given, we define the corresponding Dirichlet L-function by

An equivalent relationship had been conjectured by Euler over a hundred years earlier, in 1749, for the Dirichlet eta function ( alternating zeta function )

It is a particular Dirichlet L-function, the L-function for the alternating character of period four.

Note that if A and B are disjoint subsets of some weighted set ( U, w ), then the Dirichlet series for their ( disjoint ) union is equal to the sum of their Dirichlet series:

the sum running over all Dirichlet characters mod k. In the opposite direction we have the linear combination

where here stands for the Legendre symbol ( a / p ), and the sum is taken over residue classes modulo p. More generally, given a Dirichlet character χ mod n, the Gauss sum mod n associated with χ is

For the special case of the principal Dirichlet character, the Gauss sum reduces to the Ramanujan sum:

The support of the Dirichlet distribution is the set of-dimensional vectors whose entries are real numbers in the interval ; furthermore,, i. e. the sum of the coordinates is 1.

: If we define the concentration parameter as the sum of the Dirichlet parameters for each dimension, the Dirichlet distribution is uniform with a concentration parameter is K, the dimension of the distribution.

Similar techniques involving holomorphic Eisenstein series twisted by a Dirichlet character produce formulas for the number of representations of a positive integer n as a sum of two, four, or eight squares in terms of the divisors of n.

In mathematics, the Dirichlet conditions are sufficient conditions for a real-valued, periodic function f ( x ) to be equal to the sum of its Fourier series at each point where f is continuous.

The temperature distribution in the interior will then be given by the solution to the corresponding Dirichlet problem.

The sequence generated by a Dirichlet series generating function corresponding to:

Moreover, if χ is principal, then the corresponding Dirichlet L-function has a simple pole at s

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.