Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 080.26304
Autor: Erdös, Paul; Jabotinsky, Eri
Title: On sequences of integers generated by a sieving process. I, II. (In English)
Source: Nederl. Akad. Wet., Proc., Ser. A 61, 115-123, 124-128 (1958).
Review: These papers deals with a family of algorithms somewhat similar to the Sieve of Erathostenes. The algorithms depend on an initial integer \lambda and on a sequence B of integers bk (k = 1,2,...) with bk \geq 2. A family of intermediary sequences A(i) (i = 1,2,...) consisting of integers ak(i) (k = 1,2,...) is formed in the following way: A(1) is defined by ak(1) = \lambda+k. A(i+1) is obtained from A(i) by striking out all the terms of the form a1+mbi(i) (m = 0,1,...) and by renaming terms. Finally, the sequence A consisting of integers ak (k = 1,2,...) is defined by ak = a1(k). Two examples of sieves are considered. In the first example bk = k+1, in the second bk = ak. For bk = k+1 it is shown that ak = k2/\pi+O(k4/3). For bk = ak that ak ~ k log k. ak is in this case for every \lambda asymptotic to the primes, and the proof has some similarity to that of the prime number theorem. Because of the great regularity of the process compared to the Erathostenes method, the asymptotic formula for ak in this case is obtained more easily than that for the primes. A question by Viggo Brun has been answered by the authors, turning out to be a problem solvable by the method used in dealing with the case bk = k+1. A slight variant for the case bk = ak has been studied by Gardiner-Lazarus-Metropolis-Ulam (Zbl 071.27002).
Reviewer: S.Selberg
Classif.: * 11M35 Other zeta functions
Index Words: Number Theory
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