相关论文: Uniform bounds in Waring's problem over some diago…

For a natural number $k>1$, let $f_k(n)$ denote the number of distinct representations of a natural number $n$ of the form $p^k+q^k$ for primes $p,q$. We prove that, for all $k>1$, $$\limsup_{n\to\infty}f_k(n)=\infty.$$ This positively…

We establish uniform bounds for the sup-norms of modular forms of arbitrary real weight $k$ with respect to a finite index subgroup $\Gamma$ of $\mathrm{SL}_2(\mathbb{Z})$. We also prove corresponding bounds for the supremum over a compact…

In this methodological paper, we first review the classic cubic Diophantine equation $a^3 + b^3 + c^3 = d^3$, and consider the specific class of solutions $q_1^3 + q_2^3 + q_3^3 = q_4^3$ with each $q_i$ being a binary quadratic form. Next…

In this article, we consider the representation of $m$-gonal forms over $\mathbb N_0$. We show that any $m$-gonal forms over $\mathbb N_0$ of rank $\ge 5$ is almost regular and ponder the sufficiently large integers which are indeed…

In an earlier paper [4], we derived asymptotic formulas for the number of representations of zero and of large positive integers by the cubic forms in seven variables which can be written as $L_1(x_1,x_2,x_3) Q_1(x_1,x_2,x_3)+…

We study the differential uniformity of the Wan-Lidl polynomials over finite fields. A general upper bound, independent of the order of the field, is established. Additional bounds are established in settings where one of the parameters is…

In this paper, we present several explicit formulas of the sums and hyper-sums of the powers of the first (n+1)-terms of a general arithmetic sequence in terms of Stirling numbers and generalized Bernoulli polynomials.

We find upper bounds that are sharp for the number of $k$th powers inside arbitrary arithmetic progressions whose step has $O(1)$ many divisors.

We establish the existence of weak solutions of a nonlinear radiation-type boundary value problem for elliptic equation on divergence form with discontinuous leading coefficient. Quantitative estimates play a crucial role on the real…

We provide bounds on the sizes of the gaps -- defined broadly -- in the set $\{k_1\beta_1 + \ldots + k_n\beta_n \mbox{ (mod 1)} : k_i \in \mathbb Z \cap (0,Q^\frac{1}{n}]\}$ for generic $\beta_1, \ldots, \beta_n \in \mathbb R^m$ and all…

Recent progress on Vinogradov's mean value theorem has resulted in improved estimates for exponential sums of Weyl type. We apply these new estimates to obtain sharper bounds for the function $H(k)$ in the Waring--Goldbach problem. We…

We show that the compositions of positive integers may be interpreted in terms of powers of some power series, over arbitrary commutative ring. As consequences, several closed formulas for the compositions as well as for the generalized…

Let $k$ and $N$ be positive integers with $k\ge2$ even. In this paper we give general explicit upper-bounds in terms of $k$ and $N$ from which all the residual representations $\bar{\rho}_{f,\lambda}$ attached to non-CM newforms of weight…

We give an estimate of exponential sums over singular binary quintic forms in a characteristic-free form, based on the Waring decomposition of binary forms. This extends the method on our preceding result on the space of binary quartics to…

In this paper, we will investigate the solvability of the equation $x_1^k + x_2^k + \ldots + x_s^k = n$, $n\in \mathbb{Z}_{p^k}$, $x_1,...,x_s\in \mathcal{A}$, $\mathcal{A}\subseteq \mathbb{Z}_{p^k}$. We will give a upper bound of the…

Let \pi be a unitary cuspidal automorphic representation for GL(n) over a number field. We establish upper bounds on the number of Hecke eigenvalues of \pi equal to a fixed complex number. For GL(2), we also determine upper bounds on the…

In this paper, we study the set of positive integers that characterize the universality of $m$-gonal form.

The main purpose of this paper is to obtain strong laws of large numbers for arrays or weighted sums of random variables under a scenario of dependence. Namely, for triangular arrays $\{X_{n,k}, \, 1 \leqslant k \leqslant n, \, n \geqslant…

We estimate the number of integer solutions to decomposable form inequalities (both asymptotic estimates and upper bounds are provided) when the degree of the form and the number of variables are relatively prime. These estimates display…

Sums of positive integer powers have captivated the attention of mathematicians since ancient times. Over the centuries, mathematicians from diverse backgrounds have provided expressions for the sum of positive integer powers of the first…