On the number of common values of arithmetic functions \(\varphi\) and \(\sigma\) below \(x\) (Q446292)

Recently, \textit{K. Ford, F. Luca} and \textit{C. Pomerance} [Bull. Lond. Math. Soc. 42, No. 3, 478--488 (2010; Zbl 1205.11010)], proved that there exists a constant \(a>0\) such that for \(x>x_0\), the number of common values of the arithmetic functions \(\varphi\) (the Euler function) and \(\sigma\) (the sum of divisors function) is at least \(\exp((\log\log x)^a)\). In the paper under review the author proves that the above lower bound holds for any \(a>0\) provided that \(x>x(a)\) is sufficiently large (with respect to \(a\)). The method is a refinement of an approach of Konyagin to the proof of the result of Ford, Luca and Pomerance (loc. cit.), which appears as a remark at the end of their paper.