A Sieve Auxiliary Function

. In the sieve theories of Rosser-Iwaniec and DiamondHalberstam -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 \Gamma 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 deriva..