In the field of mathematical analysis, a general Dirichlet series is an infinite series that takes the form of where a_n, s are complex numbers and is a strictly increasing sequence of nonnegative real numbers that tends to infinity. A simple observation shows that an 'ordinary' Dirichlet series is obtained by substituting while a power series is obtained when \lambda_n=n.

Fundamental theorems

If a Dirichlet series is convergent at, then it is uniformly convergent in the domain and convergent for any s=\sigma+ti where. There are now three possibilities regarding the convergence of a Dirichlet series, i.e. it may converge for all, for none or for some values of s. In the latter case, there exist a \sigma_c such that the series is convergent for and divergent for. By convention, if the series converges nowhere and if the series converges everywhere on the complex plane.

Abscissa of convergence

The abscissa of convergence of a Dirichlet series can be defined as \sigma_c above. Another equivalent definition is The line is called the line of convergence. The half-plane of convergence is defined as The abscissa, line and half-plane of convergence of a Dirichlet series are analogous to radius, boundary and disk of convergence of a power series. On the line of convergence, the question of convergence remains open as in the case of power series. However, if a Dirichlet series converges and diverges at different points on the same vertical line, then this line must be the line of convergence. The proof is implicit in the definition of abscissa of convergence. An example would be the series which converges at s=-\pi i (alternating harmonic series) and diverges at s=0 (harmonic series). Thus, \sigma=0 is the line of convergence. Suppose that a Dirichlet series does not converge at s=0, then it is clear that and \sum a_n diverges. On the other hand, if a Dirichlet series converges at s=0, then and \sum a_n converges. Thus, there are two formulas to compute \sigma_c, depending on the convergence of \sum a_n which can be determined by various convergence tests. These formulas are similar to the Cauchy–Hadamard theorem for the radius of convergence of a power series. If \sum a_k is divergent, i.e., then \sigma_c is given by If \sum a_k is convergent, i.e., then \sigma_c is given by

Abscissa of absolute convergence

A Dirichlet series is absolutely convergent if the series is convergent. As usual, an absolutely convergent Dirichlet series is convergent, but the converse is not always true. If a Dirichlet series is absolutely convergent at s_0, then it is absolutely convergent for all s where. A Dirichlet series may converge absolutely for all, for no or for some values of s. In the latter case, there exist a \sigma_a such that the series converges absolutely for and converges non-absolutely for. The abscissa of absolute convergence can be defined as \sigma_a above, or equivalently as The line and half-plane of absolute convergence can be defined similarly. There are also two formulas to compute \sigma_a. If \sum |a_k| is divergent, then \sigma_a is given by If \sum |a_k| is convergent, then \sigma_a is given by In general, the abscissa of convergence does not coincide with abscissa of absolute convergence. Thus, there might be a strip between the line of convergence and absolute convergence where a Dirichlet series is conditionally convergent. The width of this strip is given by In the case where L = 0, then All the formulas provided so far still hold true for 'ordinary' Dirichlet series by substituting.

Other abscissas of convergence

It is possible to consider other abscissas of convergence for a Dirichlet series. The abscissa of bounded convergence \sigma_b is given by while the abscissa of uniform convergence \sigma_u is given by These abscissas are related to the abscissa of convergence \sigma_c and of absolute convergence \sigma_a by the formulas , and a remarkable theorem of Bohr in fact shows that for any ordinary Dirichlet series where (i.e. Dirichlet series of the form ), and Bohnenblust and Hille subsequently showed that for every number there are Dirichlet series for which A formula for the abscissa of uniform convergence \sigma_u for the general Dirichlet series is given as follows: for any N \geq 1, let, then

Analytic functions

A function represented by a Dirichlet series is analytic on the half-plane of convergence. Moreover, for

Further generalizations

A Dirichlet series can be further generalized to the multi-variable case where, k = 2, 3, 4,..., or complex variable case where , m = 1, 2, 3,...