We consider $k$-free numbers over Beatty sequences. New results are given. In particular, for a fixed irrational number $\alpha >1$ of finite type $\tau <\infty$ and any constant $\epsilon >0$, we can show that $$\sum _{\genfrac{}{}{0pt}{}{1\le n\le x}{[\alpha n+\beta ]\in {𝒬}_k}}1-\frac{x}{\zeta \left(k\right)}\ll x^{k/(2k-1)+\epsilon }+x^{1-1/(\tau +1)+\epsilon },$$ where ${𝒬}_k$ is the set of positive $k$-free integers and the implied constant depends only on $\alpha ,$$\epsilon ,$$k$...

For any positive integer $k\ge 2$, let ${\left(P_n^{\left(k\right)}\right)}_{n\ge 2-k}$ be the $k$-generalized Pell sequence which starts with $0,\cdots ,0,1$ ($k$ terms) with the linear recurrence $$P_n^{\left(k\right)}=2P_{n-1}^{\left(k\right)}+P_{n-2}^{\left(k\right)}+\cdots +P_{n-k}^{\left(k\right)}\text{for}n\ge 2.$$ Let ${\left(N_n\right)}_{n\ge 0}$ be Narayana’s sequence given by $$N_0=N_1=N_2=1\text{and}{N}_{n+3}=N_{n+2}+N_n.$$ The purpose of this paper is to determine all $k$-Pell numbers which are sums of two Narayana’s numbers. More precisely, we study the Diophantine equation $$P_p^{\left(k\right)}=N_n+N_m$$ in nonnegative integers $k$, $p$, $n$ and $m$.

We give a necessary and sufficient condition such that, for almost all s ∈ ℝ, ||nθ - s|| < ψ(n) for infinitely many n ∈ ℕ, where θ is fixed and ψ(n) is a positive, non-increasing sequence. This can be seen as a dual result to classical theorems of Khintchine and Szüsz which dealt with the situation where s is fixed and θ is random. Moreover, our result contains several earlier ones as special cases: two old theorems of Kurzweil, a theorem of Tseng and a recent...

We obtain a series of new conditional lower bounds for the modulus and the argument of the Riemann zeta function on very short segments of the critical line, based on the Riemann hypothesis. In particular, we prove that for any large fixed constant A > 1 there exist(non-effective) constants T₀(A) > 0 and c₀(A) > 0 such that the maximum of |ζ (0.5+it)| on the interval (T-h,T+h) is greater than A for any T > T₀ and h = (1/π)lnlnln{T}+c₀.