A Sieve Auxiliary Function

Abstract. 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 differential-difference equations with retarded arguments. To aid in the study of these functions, Iwaniec introduced a conjugate difference-differential 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. In this talk, 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.