If a Dirichlet series is convergent at, then it is uniformly convergent in the domain and convergent for any 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 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 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 and diverges at . Thus, is the line of convergence. Suppose that a Dirichlet series does not converge at, then it is clear that and diverges. On the other hand, if a Dirichlet series converges at, then and converges. Thus, there are two formulas to compute, depending on the convergence of 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 is divergent, i.e., then is given by If is convergent, i.e., then is given by
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, 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 such that the series converges absolutely for and converges non-absolutely for. The abscissa of absolute convergence can be defined as above, or equivalently as The line and half-plane of absolute convergence can be defined similarly. There are also two formulas to compute. If is divergent, then is given by If is convergent, then 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 is given by while the abscissa of uniform convergence is given by These abscissas are related to the abscissa of convergence and of absolute convergence by the formulas and a remarkable theorem of Bohr in fact shows that for any ordinary Dirichlet series where , and Bohnenblust and Hille subsequently showed that for every number there are Dirichlet series for which A formula for the abscissa of uniform convergence for the general Dirichlet series is given as follows: for any, let, then
