Dirichlet regularization assigns a value to a series 
by introducing the Dirichlet series
 |
If this series converges in a right half-plane and has an analytic continuation that is
regular at 
,
its Dirichlet-regularized value is defined to be 
. If 
has a pole at 0, a separately stated
finite-part prescription may instead use the constant term of its Laurent
series.
For example, taking 
gives 
and hence the Dirichlet-regularized value 
.
This is not the limit of the ordinary partial sums.
Dirichlet regularization is one type of regularized
sum; it should not be confused with the related zeta-regularized
sum of a positive sequence, which evaluates a spectral zeta function at 
.
