A set $𝒮=\{x_1,...,x_n\}$ of $n$ distinct positive integers is said to be gcd-closed if $(x_i,x_j)\in 𝒮$ for all $1\le i,j\le n$. Shaofang Hong conjectured in 2002 that for a given positive integer $t$ there is a positive integer $k\left(t\right)$ depending only on $t$, such that if $n\le k\left(t\right)$, then the power LCM matrix $\left({[x_i,x_j]}^t\right)$ defined on any gcd-closed set $𝒮=\{x_1,...,x_n\}$ is nonsingular, but for $n\ge k\left(t\right)+1$, there exists a gcd-closed set $𝒮=\{x_1,...,x_n\}$ such that the power LCM matrix $\left({[x_i,x_j]}^t\right)$ on $𝒮$ is singular. In 1996, Hong proved $k\left(1\right)=7$ and noted $k\left(t\right)\ge 7$ for all $t\ge 2$. This paper develops Hong’s method and provides a new idea to calculate...

In 1989, E. Saias established an asymptotic formula for $\Psi (x,y)=\left|\left\{n\le x:p\mid n\Rightarrow p\le y\right\}\right|$ with a very good error term, valid for $exp\left({(loglogx)}^{(5/3)+ϵ}\right)\le y\le x$, $x\ge x_0\left(ϵ\right)$, $ϵ\>0.$ We extend this result to an algebraic number field $K$ by obtaining an asymptotic formula for the analogous function ${\Psi }_K(x,y)$ with the same error term and valid in the same region. Our main objective is to compare the formulae for $\Psi (x,y)$ and ${\Psi }_K(x,y),$ and in particular to compare the second term in the two expansions.

We obtain the values concerning $(\theta ,\varphi )=limin{f}_{\left|q\right|\to \infty }\left|q\right|\Vert q\theta -\varphi \Vert$ using the algorithm by Nishioka, Shiokawa and Tamura. In application, we give the values (θ,1/2), (θ,1/a), (θ,1/√(ab(ab+4))) and so on when θ = (√(ab(ab+4)) - ab)/(2a) = [0;a,b,a,b,...].

We consider the values concerning$$ℳ(\theta ,\phi )=\underset{\left|q\right|\to \infty }{lim\; inf}\left|q\right|\left|\right|q^{\theta }-\phi \left|\right|$$where the continued fraction expansion of $\theta$ has a quasi-periodic form. In particular, we treat the cases so that each quasi-periodic form includes no constant. Furthermore, we give some general conditions satisfying $ℳ(\theta ,\phi )=0$.

We consider some applications of the singular integral equation of the second kind of Fox. Some new solutions to Fox’s integral equation are discussed in relation to number theory.

The phenomenon of anomaly small error terms in the lattice point problem is considered in detail in two dimensions. For irrational polygons the errors are expressed in terms of diophantine properties of the side slopes. As a result, for the $t$-dilatation, $t\to \infty$, of certain classes of irrational polygons the error terms are bounded as $\ll \ell n^{q}t$ with some $q\>0$, or as $\ll t^{ϵ}$ with arbitrarily small $ϵ\>0$.