相关论文: Linear relations among asymptotic frequencies in c…

Given a positive definite binary quadratic form f, let r(n) = |{(x,y): f(x,y)=n}| denote its representation function. In this paper we study linear correlations of these functions. For example, if r_1, ..., r_k are representation functions,…

Suppose that $ m\equiv 1\mod 4 $ is a prime and that $ n\equiv 3\mod 4 $ is a primitive root modulo $ m $. In this paper we obtain a relation between the class number of the imaginary quadratic field $ \Q(\sqrt{-nm}) $ and the digits of the…

We give conditions on sequences of positive algebraic numbers $\{a_{n,j}\}_{n=1}^\infty$, $j=1,\dots ,M$ and number field $\mathbb K$ to ensure that the numbers defined by the continued fractions $[0;a_{1,j},a_{2,j},\dots ]$, $j=1,\dots ,M$…

We initiate the study of the sets $H(c)$, $0<c<1$, of real $x$ for which the sequence $(kx)_{k\geq1}$ (viewed mod 1) consistently hits the interval $[0,c)$ at least as often as expected (i. e., with frequency $\geq c$). More formally, \[…

In this paper, we investigate the monotone property of the continued fractions $G(m,\lambda)$ as a function of $m$ and $\lambda$. In particular, we obtain new inequality for the relative continued fractions.

Asymptotic formulae are established for the number of natural numbers $m$ with largest square-free divisor not exceeding $m^{\vartheta}$, for any fixed positive parameter $\vartheta$. Related counting functions are also considered.

Let $q$ be an odd prime power and let $X(m,q)$ be the set of symmetric bilinear forms on an $m$-dimensional vector space over $\mathbb{F}_q$. The partition of $X(m,q)$ induced by the action of the general linear group gives rise to a…

Using general local conservations laws we derive dispersion relations for edge modes in a slab of electron liquid confined by a symmetric potential. The dispersion relations are exact up to $\lambda^{2} q^{2}$, where $q$ is a wave vector…

Let $I(b,d,k)$ be the subseries of the harmonic series keeping the integers having exactly $k$ occurrences of the digit $d$ in base $b$. We prove the existence of an asymptotic expansion to all orders in descending powers of $b$, for fixed…

Let $f(x)$ be a monic polynomial over $\mathbb{Q}$ with complex roots $\alpha_1,\dots,\alpha_n$. Linear relations among them and $1$ over $\mathbb{Q}$ play an important role when we study the distribution of roots modulo a prime. We study…

In "Random complex fewnomials, I," B. Shiffman and S. Zelditch determine the limiting formula as N goes to infinity of the (normalized) expected distribution of complex zeros of a system of k random n-nomials in m variables where the…

The connection between a Taylor series and a continued-fraction involves a nonlinear relation between the Taylor coefficients $\{ a_n \}$ and the continued-fraction coefficients $\{ b_n \}$. In many instances it turns out that this…

Let $q$ be an integer. A $D(q)$-$m$-tuple is a set of $m$ distinct positive integers ${a_1, a_2, . . . , a_m}$ such that $a_ia_j + q$ is a perfect square for all $1 \leq i < j \leq m$. By counting integer solutions $x \in [1, b]$ of…

Rational approximations to a square root $\sqrt{k}$ can be produced by iterating the transformation $f(x) = (dx+k)/(x+d)$ starting from $\infty$ for any positive integer $d$. We show that these approximations coincide infinitely often with…

Let $\mathscr{Q}(m,q)$ and $\mathscr{S}(m,q)$ be the sets of quadratic forms and symmetric bilinear forms on an $m$-dimensional vector space over $\mathbb{F}_q$, respectively. The orbits of $\mathscr{Q}(m,q)$ and $\mathscr{S}(m,q)$ under a…

Let $M_{d,n}(q)$ denote the number of monic irreducible polynomials in $\mathbb{F}_q[x_1, x_2, \ldots , x_n]$ of degree $d$. We show that for a fixed degree $d$, the sequence $M_{d,n}(q)$ converges $q$-adically to an explicitly determined…

Harmonic numbers $H_k=\sum_{0<j\le k}1/j (k=0,1,2,...)$ arise naturally in many fields of mathematics. In this paper we initiate the study of congruences involving both harmonic numbers and Lucas sequences. One of our three theorems is as…

For positive integers $q$, Dirichlet's theorem states that there are infinitely many primes in each reduced residue class modulo $q$. A stronger form of the theorem states that the primes are equidistributed among the $\varphi(q)$ reduced…

Four different relations between the Legendre nodes and weights are presented which, unlike the circle and trapezoid theorems for Gauss-Legendre quadrature, hold uniformly in the whole interval $(-1,1)$. These properties are supported by…

We have for positive integers $n$, $k$ and finite field $\mathbb{F}_q$, $c(n,k,q)$, as the number of simultaneous similarity classes of $k$-tuples of commuting $n\times n$ matrices over the $\mathbb{F}_q$. In this paper, it has been shown…