Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 173.03903
Autor: Erdös, Pál
Title: Some remarks on the iterates of the \phi and \sigma functions (In English)
Source: Colloq. Math. 17, 195-202 (1967).
Review: Put \phi1 = \phi (n), \phi k(n) = \phi1 (\phik-1 (n)) and denote by N\phi (k,\alpha ,x) the number of integers n \leq x for which \phik (n) > \alpha n. The author proves that for every t, \alpha < 1/2 , \epsilon > 0, and x > x0(\alpha , t, \epsilon) the inequalities hold x(log log x)t/ log x < N\phi (2,\alpha, x) < x(log x)\epsilon / log x; further, for every \alpha > 0, \epsilon > 0 and x < x0 (a,\epsilon)
N\phi (3,\alpha ,x) < x(log x)\epsilon /(log x)2. A similar theorem for the \sigma function is given without proof and many related question are discussed.
Reviewer: A.Schinzel
Classif.: * 11A25 Arithmetic functions, etc.
11N64 Characterization of arithmetic functions
Index Words: number theory
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