Common values of the arithmetic functions phi and sigma
Number Theory
2014-02-26 v2
Abstract
We show that the equation phi(a)=\sigma(b) has infinitely many solutions, where phi is Euler's totient function and sigma is the sum-of-divisors function. This proves a 50-year old conjecture of Erdos. Moreover, we show that there are infinitely many integers n such that phi(a)=n and sigma(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 phi at p, and on a result of Heath-Brown connecting the possible existence of Siegel zeros with the distribution of twin primes.
Cite
@article{arxiv.0906.3380,
title = {Common values of the arithmetic functions phi and sigma},
author = {Kevin Ford and Florian Luca and Carl Pomerance},
journal= {arXiv preprint arXiv:0906.3380},
year = {2014}
}
Comments
v2, Aug. 2009. Small corrections and changes