An improved upper bound for the distribution of iterated Euler totient functions

2025/06/30 by Pei Gao, Qiyu Yang, Gao, Pei +1
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.2506.23882

Abstract

Let ϕ(n) be the Euler totient function and ϕ_k(n) its k-fold iterate. In this note, we improve the upper bound for the number of positive n\leqslant x such that ϕ_k+1(n)\geqslant cn. Comparing with the upper bound which was obtained from Pollack's asymptotic formula of the summation of ϕ_k+1(n) for n\leqslant x, we have successfully increased the denominator exponent of the main term of the upper bound from k to k+1.

Related

• The number of preimages of iterates of ϕ and σ
• Distribution of values of general Euler totient function
• A Generalisation of Euler Totient Function
• The distribution functions of σ(n)/n and n/ϕ(n), II
• Sums related to Euler's totient function