Bulletin of the Australian Mathematical Society

Research Article

Laurent expansion of Dirichlet series

U. Balakrishnana1

a1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay – 5, India.

Abstract

Let xs3008anxs3009 be an increasing sequence of real numbers and xs3008bn a sequence of positive real numbers. We deal here with the Dirichlet series S0004972700003919_inline1 and its Laurent expansion at the abscissa of convergence, λ, say. When an and bn behave like

S0004972700003919_eqnU1

as N → ∞, where P2(x) is a certain polynomial, we obtain the Laurent expansion of f (s) at s = λ, namely

S0004972700003919_eqnU2

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.

(Received July 26 1985)