The finite sum is a discrete version of a definite integral, the discrete version of the derivative. A discrete version of the partial integration formula
A sequence of functions defined on a region converge uniformely to a
function if
.
It follows from the Jensen-Cahen Theorem that for every region in the cone , the series is uniformly convergent as well as any of its derivatives .
The abscissa of simple convergence of a Dirichlet series
is
converges for all
.
The abscissa of absolute convergence of is
converges absolutely for all
.
Example. The Dirichlet eta function
has the abscissa of
convergence
and the absolute abscissa of convergence
.
Assume a Dirichlet series is not convergent for . In other words, the series does not converge. The following formula generalizes the formula for the radius of convergence for Taylor series, where and where the radius of convergence is related with the abscissa of convergence by .
Similarly, there is a formula for the abscissa of absolute convergence:
,
where
.
Cahen's formula links the growth of the random walk
with the convergence properties of the zeta function
.
Example: has and .
Source: [1].