Let Γ ⊂ ℚ * be a finitely generated subgroup and let p be a prime such that the reduction group Γₚ is a well defined subgroup of the multiplicative group ₚ*. We prove an asymptotic formula for the average of the number of primes p ≤ x for which [ₚ*:Γₚ] = m. The average is taken over all finitely generated subgroups $\Gamma =⟨a₁,...,a_r⟩\subset \mathbb{Q}*$, with $a_i\in \mathbb{Z}$ and $a_i\le T_i$, with a range of uniformity $T_i>exp\left(4{\left(logxloglogx\right)}^{1/2}\right)$ for every i = 1,...,r. We also prove an asymptotic formula for the mean square of the error terms in the asymptotic formula with a similar...

We study values of the Euler function φ(n) taken on binary palindromes of even length. In particular, if ${ℬ}_{2\ell }$ denotes the set of binary palindromes with precisely 2ℓ binary digits, we derive an asymptotic formula for the average value of the Euler function on ${ℬ}_{2\ell }$.