COMMON VALUES OF THE ARITHMETIC FUNCTIONS φ AND σ
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BibTeX
@MISC{Ford_commonvalues,
author = {Kevin Ford and Florian Luca and Carl Pomerance},
title = {COMMON VALUES OF THE ARITHMETIC FUNCTIONS φ AND σ},
year = {}
}
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Abstract
ABSTRACT. We show that the equation φ(a) = σ(b) has infinitely many solutions, where φ is Euler’s totient function and σ is the sum-of-divisors function. This proves a 50-year old conjecture of Erdős. Moreover, we show that there are infinitely many integers n such that φ(a) = n and σ(b) = n each have more than n c solutions, for some c> 0. The proofs rely on the recent work of the first two authors and Konyagin on the distribution of primes p for which a given prime divides some iterate of φ at p, and on a result of Heath-Brown connecting the possible existence of Siegel zeros with the distribution of twin primes. 1.
