Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  512.10037
Autor:  Erdös, Paul; Sárközy, András
Title:  Some asymptotic formulas on generalized divisor functions. IV. (In English)
Source:  Stud. Sci. Math. Hung. 15, 467-479 (1980).
Review:  The paper under review is a sequel to three papers of the authors [Part I, Studies in pure Mathematics, Mem. of P. Turán, 165-179 (1983), II. J. Number Theory 15, 115-136 (1982; Zbl 488.10043) and III. Acta Arith. 41, 395-411 (1982; Zbl 492.10037)]. For a given sequence A: a1 < a2 < ... of positive integers define NA(x) = suma in A,a \leq x1, fA(x) = suma in A,a \leq x 1/a , the divisor function \tauA(n) = suma \leq A,a|n1, and its maximum DA(x) = max1 \leq n \leq x\tauA(n). The authors are interested in ``large'' values of DA(x) compared with fA(x), the normal order of \tauA(n). For example, in III the authors showed that for all \Omega > 0, and for x > X0(\Omega) DA(x)/fA(x) > \Omega, if fA(X) > (log log x)20. Now it is shown that for any \Omega > 1 there exist constants c3(\Omega), X1(\Omega) such that DA(x)/fA(x) > \Omega, if x > X1, if fA(x) > c3 and if the interval [x1-{fA(x)-1/3},x] has void intersection with A. As the authors show, this theorem does not remain true, if 1-{fA(x)}-1/3 is replaced by 1-c6-fA(x).
Reviewer:  W.Schwarz
Classif.:  * 11N37 Asymptotic results on arithmetic functions
Keywords:  divisor functions; sets of integers; large values
Citations:  Zbl.488.10043; Zbl.492.10037

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