On the Riemann zeta-function and related problems
by
A. Sankaranarayanan
Tata Institute of Fundamental Research, Mumbai 400 005, India

Let F(s) be a Dirichlet series which is a quotient of some products of the translates of the Riemann zeta-function, that is

F(s)= \equiv æ è P(s) Õ \alpha in N   \zeta(s+\alpha) ö ø   æ è Õ \beta in D   \zeta(s + \beta) ö ø -1   \equiv F1(s)(F2(s))-1,

where N and D are finite sets of complex numbers \alpha and \beta, respectively (need not be distinct) and P(s) is a Dirichlet polynomial. Suppose that F(s) is a non-terminating Dirichlet series. Under certain conditions on N and D, we investigate the singularities of F(s). Precisely we prove that there are infinitely many poles p₁+ip₂ in Im(s) > C for any fixed C > 0. Also under certain conditions on the coefficients of F(s) (which is fairly satisfied for many F(s)), we study the gaps between the ordinates of the consecutive poles of F(s).

Date received: March 22, 1999