On the analytic continuation of a certain Dirichlet series

A Dirichlet series associated with a positive definite form of degree δ in n variables is defined by D F (s,p,α) = ∑ α∈Z n −{0} F(α) −s e(ρF(α)+〈α, α〉) where ϱ ∈ Q , α ∈ Q n, 〈x, y〉 = x1y1 + ⋯ + xnyn, e(a) = exp (2πia) for a ∈ R , and s = σ + ti is a complex number. The author proves that: (1) DF(s, ϱ, α) has analytic continuation into the whole s-plane, (2) DF(s, ϱ, α), ϱ ≠ 0, is a meromorphic function with at most a simple pole at s = n δ . The residue at s = n δ is given explicitly. (3) ϱ = 0, α ∉ Z n, DF(s, 0, α) is analytic for α>, n (δ − 1) .