Let $\pi $ be an automorphic irreducible cuspidal representation of $\mathrm{GL}_{m}$ over $\mathbb {Q}$ . Denoted by $\lambda _{\pi }(n)$ the nth coefficient in the Dirichlet series expansion of $L(s,\pi )$ associated with $\pi $ . Let $\pi _{1}$ be an automorphic irreducible cuspidal representation of $\mathrm{SL}(2,\mathbb {Z})$ . Denoted by $\lambda _{\pi _{1}\times \pi _{1}}(n)$ the nth coefficient in the Dirichlet series expansion of $L(s,\pi _{1}\times \pi _{1})$ associated with $\pi _{1}\times \pi _{1}$ . In this paper, we study the cancellations of $\lambda _{\pi }(n)$ and $\lambda _{\pi _{1}\times \pi _{1}}(n)$ over Beatty sequences.

We consider a concrete family of $2$ -towers $(\mathbb {Q}(x_n))_n$ of totally real algebraic numbers for which we prove that, for each $n$ , $\mathbb {Z}[x_n]$ is the ring of integers of $\mathbb {Q}(x_n)$ if and only if the constant term of the minimal polynomial of $x_n$ is square-free. We apply our characterization to produce new examples of monogenic number fields, which can be of arbitrary large degree under the ABC-Conjecture.

We give a corrected version of our previous lower bound on the value set of binomials (Canad. Math. Bull., v.63, 2020, 187–196). The other results are not affected.

Let $k\geqslant 1$ be a natural number and $\omega _k(n)$ denote the number of distinct prime factors of a natural number n with multiplicity k. We estimate the first and second moments of the functions $\omega _k$ with $k\geqslant 1$ . Moreover, we prove that the function $\omega _1(n)$ has normal order $\log \log n$ and the function $(\omega _1(n)-\log \log n)/\sqrt {\log \log n}$ has a normal distribution. Finally, we prove that the functions $\omega _k(n)$ with $k\geqslant 2$ do not have normal order $F(n)$ for any nondecreasing nonnegative function F.