Laurent expansion of Dirichlet series

Let 〈an〉 be an increasing sequence of real numbers and 〈bn a sequence of positive real numbers. We deal here with the Dirichlet series and its Laurent expansion at the abscissa of convergence, λ, say. When an and bn behave like as N → ∞, where P2(x) is a certain polynomial, we obtain the Laurent expansion of f (s) at s = λ, namely where P1(x) is a polynomial connected with P2(x) above. Also, the connection between P1 and P2 is made intuitively transparent in the proof.