General Dirichlet series

In the field of mathematical analysis, a general Dirichlet series is an infinite series that takes the form of
where formula_2, formula_3 are complex numbers and formula_4 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 formula_6 while a power series
is obtained when formula_8.
Fundamental theorems.
If a Dirichlet series is convergent at formula_9, then it is uniformly convergent in the domain
and convergent for any formula_11 where formula_12.
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 formula_13 such that the series is convergent for formula_14 and divergent for formula_15. By convention, formula_16 if the series converges nowhere and formula_17 if the series converges everywhere on the complex plane.
Abscissa of convergence.
The abscissa of convergence of a Dirichlet series can be defined as formula_13 above. Another equivalent definition is
The line formula_20 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 formula_23 (alternating harmonic series) and diverges at formula_24 (harmonic series). Thus, formula_25 is the line of convergence.
Suppose that a Dirichlet series does not converge at formula_24, then it is clear that formula_27 and formula_28 diverges. On the other hand, if a Dirichlet series converges at formula_24, then formula_30 and formula_28 converges. Thus, there are two formulas to compute formula_13, depending on the convergence of formula_28 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 formula_34 is divergent, i.e. formula_27, then formula_13 is given by
If formula_34 is convergent, i.e. formula_30, then formula_13 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 formula_43, then it is absolutely convergent for all "s" where formula_44. A Dirichlet series may converge absolutely for all, for no or for some values of "s". In the latter case, there exist a formula_45 such that the series converges absolutely for formula_46 and converges non-absolutely for formula_47.
The abscissa of absolute convergence can be defined as formula_45 above, or equivalently as
The line and half-plane of absolute convergence can be defined similarly. There are also two formulas to compute formula_45.
If formula_51 is divergent, then formula_45 is given by
If formula_51 is convergent, then formula_45 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 formula_59.
Other abscissas of convergence.
It is possible to consider other abscissas of convergence for a Dirichlet series. The abscissa of bounded convergence formula_60 is given by
formula_61
while the abscissa of uniform convergence formula_62 is given by
formula_63
These abscissas are related to the abscissa of convergence formula_13 and of absolute convergence formula_45 by the formulas
formula_66,
and a remarkable theorem of Bohr in fact shows that for any ordinary Dirichlet series where formula_67 (i.e. Dirichlet series of the form formula_68) , formula_69 and formula_70 Bohnenblust and Hille subsequently showed that for every number formula_71 there are Dirichlet series formula_68 for which formula_73
A formula for the abscissa of uniform convergence formula_62 for the general Dirichlet series formula_75 is given as follows: for any formula_76, let formula_77, then formula_78
Analytic functions.
A function represented by a Dirichlet series
is analytic on the half-plane of convergence. Moreover, for formula_80
Further generalizations.
A Dirichlet series can be further generalized to the multi-variable case where formula_82, "k" = 2, 3, 4..., or complex variable case where formula_83, "m" = 1, 2, 3...