On the prime divisors of the Euler Phi and the sum
of divisors functions

Abstract. The following assertion is proved. Given an arbitrary constant \(\lambda >2\), let \(x_1=\log x,\ x_{k+1}=\log x_k\ (k=1,2,\dots)\), \(\varphi(n)\) be Euler's totient, and \(\sigma(n)\) the sum of divisors function. Let \(I_x=\left[\displaystyle{{{\lambda x_2}\over{x_3}}}, x_2\right]\), \(Q_1,Q_2\in I_x\) be primes, $$ E_{Q_1,Q_2}(x):=\#\{n\leq x\ |\ Q_1\nmid \varphi(n),\ Q_2\nmid \varphi(n+1)\}. $$ Then, uniformly for \(Q_1,Q_2\in I_x\), $$ {{1}\over x}E_{Q_1,Q_2}(x)=(1+o_x(1)){{B}\over 2}\kappa _1\kappa _2, $$ where \(\kappa_j=\exp\left(-\displaystyle{{{x_2}\over{Q_j-1}}}\right)\) \((j=1,2)\), \(B\) is a given constant. Some other assertions are formulated without proof.