For k = 1,2,... let $H_k$ denote the harmonic number ${\sum }_{j=1}^{k}1/j$. In this paper we establish some new congruences involving harmonic numbers. For example, we show that for any prime p > 3 we have ${\sum }_{k=1}^{p-1}\left(H_k\right)/\left(k{2}^k\right)\equiv 7/24p{B}_{p-3}\left(modp²\right)$, ${\sum }_{k=1}^{p-1}\left(H_{k,2}\right)/\left(k{2}^k\right)\equiv -3/8B_{p-3}\left(modp\right)$, and ${\sum }_{k=1}^{p-1}\left(H{²}_{k,2n}\right)/\left(k^{2n}\right)\equiv (\left(\genfrac{}{}{0pt}{}{6n+1}{2n-1}\right)+n)/(6n+1)p{B}_{p-1-6n}\left(modp²\right)$ for any positive integer n < (p-1)/6, where B₀,B₁,B₂,... are Bernoulli numbers, and $H_{k,m}:={\sum }_{j=1}^{k}1/\left(j^m\right)$.

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...