A Sieve Auxiliary Function

Bradley, David M. (1996) A Sieve Auxiliary Function. [Preprint]

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      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 \cite{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.

      Item Type: Preprint
      Additional Information: pubdom FALSE
      Subjects: UNSPECIFIED
      Faculty: UNSPECIFIED
      Depositing User: Users 1 not found.
      Date Deposited: 24 Nov 2003
      Last Modified: 21 Apr 2010 11:13

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