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On the range of Carmichael's universal-exponent function

Florian LucaCarl Pomerance — 2014

Acta Arithmetica

Let λ denote Carmichael’s function, so λ(n) is the universal exponent for the multiplicative group modulo n. It is closely related to Euler’s φ-function, but we show here that the image of λ is much denser than the image of φ. In particular the number of λ-values to x exceeds x / ( l o g x ) . 36 for all large x, while for φ it is equal to x / ( l o g x ) 1 + o ( 1 ) , an old result of Erdős. We also improve on an earlier result of the first author and Friedlander giving an upper bound for the distribution of λ-values.

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

Florian LucaCarl Pomerance — 2002

Colloquium Mathematicae

Let σ(n) denote the sum of positive divisors of the integer n, and let ϕ denote Euler's function, that is, ϕ(n) is the number of integers in the interval [1,n] that are relatively prime to n. It has been conjectured by Mąkowski and Schinzel that σ(ϕ(n))/n ≥ 1/2 for all n. We show that σ(ϕ(n))/n → ∞ on a set of numbers n of asymptotic density 1. In addition, we study the average order of σ(ϕ(n))/n as well as its range. We use similar methods to prove a conjecture of Erdős that ϕ(n-ϕ(n)) < ϕ(n)...

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