On the iterates of Euler's function

Abstract

Asymptotic representations are given for the three sums

On some problems of Mąkowski–Schinzel and Erdős concerning the arithmetical functions φ and σ

“…We remark that Warlimont (see [22]) has recently shown a result similar to part (iii) of Theorem 1 for the function φ(n)/φ(φ(n)).…”

Two remarks on iterates of Euler’s totient function

“…Estimates similar to those considered in Theorem 1.1 were established by Warlimont [17] in the case k = 1; see the remarks at the end of Section 2.…”

“…Warlimont [17] calculated asymptotic formulas for the partial sums of ϕ/ϕ 2 , ϕ 2 /ϕ and log ϕ ϕ2 . There is no difficulty in calculating corresponding formulas for the partial sums of ϕ k /ϕ k+1 , ϕ k+1 /ϕ k , or log ϕ k ϕ k+1 , for any fixed k, by the methods employed in the proof of Theorem 1.1.…”

Compositions with the euler and carmichael functions

“…Since the inequalities 1 ≥ ϕ(λ(n)) λ(n) 1 log 2 λ(n) ≥ 1 log 2 n hold for all n, and the estimate λ(n) = n exp(− (1 + o(1)) log 2 n log 3 n) (49) holds for almost all n (see [19]), it follows that ϕ(λ(n)) = n exp(− (1 + o(1)) log 2 n log 3 n)…”

“…For a variety of other results with a similar flavor, we refer the reader to [3,4,5,6,8,10,12,13,21,22,23,24,26,33,34,40,42,45,49] and the references contained therein.…”