Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 352.10027
Autor: Diamond, Harold G.; Erdös, Paul
Title: A measure of the nonmonotonicity of the Euler phi function. (In English)
Source: Pac. J. Math. 77, 83-101 (1978).
Review: Let f be a real valued arithmetic function satisfying limn ––> oo f(n) = +oo. Define another arithmetic functions F = Ff by setting Ff(n) = \# {j < n: f(j) \geq f(n)}+\# {j > n: f(j) \leq f(n)}. The size of the values assumed by the function F provides a measure of the nonmonotonicity of f. In particular, F is identically zero if an only if f is strictly increasing. In the present article f = \phi , Euler's functions and F\phi is written as F. It is shown that F(n)/n is asymptotically represented as h(\phi(n)/n), where h is a certain convex function. Using this representation it is shown that F(n)/n has a distribution function. The functions maxn \leq x F(n) and maxn > x F(n) are studied and conditions on \phi(n)/n are found which lead to large and small values of F(n)/n.
Classif.: * 11K65 Arithmetic functions (probabilistic number theory)
11N37 Asymptotic results on arithmetic functions
11A25 Arithmetic functions, etc.
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