Abstract
Our aim in this note is to analyse the differential difference equations underlying sieves of dimension κ > 1. A heuristic version of such an analysis together with some valuable numerical information was given by Iwaniec, van de Lune and te Riele [5] (see also te Riele [7]) and what we seek to do here, in effect, is to justify the conclusions of [5]. It has been shown elsewhere (in [2]) how to construct sieves of dimension κ > 1 on the basis of such information. In this connection we acknowledge also our indebtedness to the important thesis of Rawsthorne [6].
All Three authors acknowlege with gratitude support from the National Science Foundation.
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References
N. C. Ankeny and H. Onishi, The general sieve, Acta Arith. 10(1964/65), 31–62.
H. Diamond and H. Halberstam, The Combinatorial Sieve, to appear in the Proceedings of the Math. Science Conference on Number Theory 1983, Springer Lecture Notes 1985.
H. Halberstam and H. -E. Richert, Sieve Methods, Academic Press, 1974.
H. Iwaniec, Rosser’s Sieve, Acta Arith. 36 (1980), 171–202.
H. Iwaniec, J. van de Lune and H. J. J. te Riele, The limits of Buchstab’s iteration sieve, Indag. Math. Proc. A 83(4), (1980).
D. Rawsthorne, Improvements in the small sieve estimate of Selberg by iteration, Ph.D. thesis, University of Illinois, 1980.
H. J. J. te Riele, Numerical solution of two coupled non linear equations related to the limits of Buchstab’s iteration sieve, Afeding Numerieke Wiskunde, 86. Math. Centrum, Amsterdam, 1980, 15 pp.
R. N. Wilson, Concerning the number of mutually orthogonal latin squares, Discrete Math. 9 (1974), 181–198
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© 1987 Birkhäuser Boston
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Diamond, H., Halberstam, H., Richert, HE. (1987). Differential Difference Equations Associated with Sieves. In: Adolphson, A.C., Conrey, J.B., Ghosh, A., Yager, R.I. (eds) Analytic Number Theory and Diophantine Problems. Progress in Mathematics, vol 70. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-4816-3_6
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