Analytic continuation and asymptotics of Dirichlet series with partitions

The analytic continuation of a family of Dirichlet series whose coefficients are partition functions having parts in a finite set is established. The singularities arising from this continuation are shown to be finitely many simple poles, where the residues are computable positive rational numbers. Then we give several applications. First, special values of the continuation are studied in terms of algebraic combinations of the Hurwitz zeta function. Second, a transformation formula is obtained for exponentially decaying series involving these partition functions using special values of the continuation at negative integers. Finally, asymptotic behavior and orthogonality of certain convolutions with the normalization of these partition functions are investigated in the mean value sense with precise error terms.