A Sieve Auxiliary Function

In the sieve theories of Rosser-Iwaniec and Diamond-Halberstam-Richert, the upper and lower bound sieve functions (F and f, respectively) satisfy a coupled system of diierential-diierence equations with retarded arguments. To aid in the study of these functions, Iwaniec introduced a conjugate diierence-diierential equation with an advanced argument, and gave a solution, q, which is analytic in the right half-plane. The analysis of the bounding sieve functions, F and f, is facilitated by an adjoint integral inner-product relation which links the local behaviour of F ? f with that of the sieve auxiliary function, q. In addition, q plays a fundamental role in determining the sieving limit of the combinatorial sieve, and hence in determining the boundary conditions of the sieve functions, F and f. The sieve auxiliary function, q, has been tabulated previously, but these data were not supported by numerical analysis, due to the prohibitive presence of high-order partial derivatives arising from the numerical quadrature methods used 15, 17]. In this paper, we develop additional representations of q. Certain of these representations are amenable to detailed error analysis. We provide this error analysis, and as a consequence, we indicate how q-values guaranteed to at least seven decimal places can be tabulated.