Paul Erdos and Carl Pomerance have proofs on an asymptotic upper bound on the number of preimages of Euler's totient function and the sum-of-divisors functions . In this paper, we will extend the upper bound to the number of preimages of iterates of and . Using these new asymptotic upper bounds, a conjecture in de Koninck and Kátai's paper, “On the uniform distribution of certain sequences involving the Euler totient function and the sum of divisors function” is now proven and many corollaries follow from their proven conjecture.

中文翻译：

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ψ 和 σ 迭代的原像数量

Paul Erdos 和 Carl Pomerance 证明了欧拉 totient 函数和除数函数的原像数量的渐近上限。在本文中，我们将把上限扩展到 和 迭代的原像数量。使用这些新的渐近上限，de Koninck 和 Kátai 的论文“关于涉及欧拉 totient 函数和除数函数之和的某些序列的均匀分布”中的猜想现已得到证明，并且许多推论都来自他们已证明的猜想。