Dirichlet series associated with polynomials and applications

We consider the possibility of the analytic continuation of the Dirichlet series SP;Z(s) associated with a polynomial P(x) and auxiliary series Z(s). In fact, we derive a certain criterion for the analytic continuation for some class of polynomials and give examples such that SP;Z(s) cannot be continued meromorphically to the whole plane C. We also study the asymptotic behaviors of the sum MP(x)=∑P(n1,…,nk)⩽xΛ(n1)⋯Λ(nk) considered first by M. Peter. Generalizations of this sum are also considered.