Dirichlet series without order term

is there a name in use for Dirichlet series without the order term, analogously to Laurent or Puiseux polynomials? Is there work known about such expressions?

$D(s) = \sum_{0<n<N}a_n/n^s$

The question came up when implementing Dirichlet series in a computer algebra system.

• Could you please clarify what you mean by "Dirichlet series without the order term"? Do you mean series of the shape $\sum_{n=-\infty}^\infty a_n/n^s$? Apr 6, 2015 at 7:55
They arise in many places in analytic number theory. For example, in approximate functional equations of $L$-functions.