General Dirichlet series

Fundamental theorems

If a Dirichlet series is convergent at s 0 = σ 0 + t 0 i , then it is uniformly convergent in the domain

and convergent for any s = σ + t i where σ > σ 0 .

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 σ c such that the series is convergent for σ > σ c and divergent for σ < σ c . By convention, σ c = ∞ if the series converges nowhere and σ c = − ∞ if the series converges everywhere on the complex plane.

Abscissa of convergence

The abscissa of convergence of a Dirichlet series can be defined as σ c above. Another equivalent definition is

The line σ = σ c 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 = − π i (alternating harmonic series) and diverges at s = 0 (harmonic series). Thus, σ = 0 is the line of convergence.

Suppose that a Dirichlet series does not converge at s = 0 , then it is clear that σ c ≥ 0 and ∑ a n diverges. On the other hand, if a Dirichlet series converges at s = 0 , then σ c ≤ 0 and ∑ a n converges. Thus, there are two formulas to compute σ c , depending on the convergence of ∑ 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 ∑ a k is divergent, i.e. σ c ≥ 0 , then σ c is given by

If ∑ a k is convergent, i.e. σ c ≤ 0 , then σ 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 Re ( s ) > Re ( s 0 ) . A Dirichlet series may converge absolutely for all, for no or for some values of s. In the latter case, there exist a σ a such that the series converges absolutely for σ > σ a and converges non-absolutely for σ < σ a .

The abscissa of absolute convergence can be defined as σ a above, or equivalently as

The line and half-plane of absolute convergence can be defined similarly. There are also two formulas to compute σ a .

If ∑ | a k | is divergent, then σ a is given by

If ∑ | a k | is convergent, then σ 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 λ n = log ⁡ n .

Analytic functions

A function represented by a Dirichlet series

is analytic on the half-plane of convergence. Moreover, for k = 1 , 2 , 3 , . . .

Further generalizations

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