The zeta-regularized sum of a positive sequence 
is obtained from the analytic
continuation of its spectral zeta function
 |
If 
has an analytic continuation that is regular
at 
,
the zeta-regularized value of 
is defined to be 
(Elizalde 1995).
Although the notation is not standardized, the continuation value 
records the prescription directly. Among the
decorated sums used as alternatives, Lipnowski and Stern
(2018) write 
.
A notation such as 
can likewise indicate zeta regularization when defined in context. For an operator 
, the corresponding zeta trace is written
and the related zeta-regularized determinant is written 
(Elizalde 2007).
For 
,
this prescription assigns 
to 
because
 |
This assignment is a statement about regularization, not convergence of the ordinary partial sums. Zeta-regularized sums
are related to, but distinct from zeta-regularized
products, which use 
.
