A remark on a theorem of the Goldbach Waring type

On the Waring-Goldbach problem on average

Let $s$, $\ell$ be two integers such that $2\le s\le \ell-1$, $\ell\ge 3$. We prove that a suitable asymptotic formula for the average number of representations of integers $n=\sum_{i=1}^{s} p_{i}^{\ell}$, where $p_i$, $i=1,\dotsc,s$, are prime numbers, holds in short intervals.

华林-哥德巴赫问题的一个密度版本

A density version of Waring-Goldbach problem

该研究探讨华林-哥德巴赫问题的密度版本，证明当下密度大于1-1/2k的素数子集满足条件时，充分大的同余自然数可表为该子集素数k次幂的s项和，推进了加性数论相关问题研究。

关于混合幂的华林-哥德巴赫问题

On Waring--Goldbach problem with mixed powers

该研究证明了充分大奇数可表为7个分别为2到8次幂的素数之和，改进了蔡与穆2015年的相关成果。

一次平方和更高次幂的瓦林-戈尔德巴奇问题

Waring-Goldbach problems for one square and higher powers

本文研究了用一个平方和更高次幂表示足够大的整数的问题，证明了奇数和偶数均可表示为素数的平方、四次幂和五次幂的组合。

关于一个素数平方与十七个素数五次幂的Waring-Goldbach问题

On Waring-Goldbach problem for one square and seventeen fifth powers of primes

本文证明每个充分大的偶数可表示为一个素数平方与十七个素数五次幂之和，改进了Brüdern和Kawada的先前结果。

Exponential sums over primes in short intervals and an application to the Waring--Goldbach problem

Let $Λ(n)$ be the von Mangoldt function, $x$ real and $2\leq y \leq x$. This paper improves the estimate on the exponential sum over primes in short intervals \[ S_k(x,y;α) = \sum_{x< n \leq x+y} Λ(n) e\left( n^k α\right) \] when $k\geq 3$ for $α$ in the minor arcs. And then combined with the Hardy--Littlewood circle method, this enables us to investigate the Waring--Goldbach problem of representing a positive integer $n$ as the sum of $s$ $k$th powers of almost equal prime numbers, which improves the results in Wei and Wooley [12].

关于Piatetski-Shapiro素数的三次Waring-Goldbach问题

Cubic Waring-Goldbach problem with Piatetski-Shapiro primes

本文研究Piatetski-Shapiro素数的三次Waring-Goldbach问题，证明了当γ∈(317/320,1)时，每个充分大的奇数可表示为九个形如[n^{1/γ}]的素数立方之和，改进了Akbal和Güloğlu的已有结果。

On the ergodic Waring--Goldbach problem

We prove an asymptotic formula for the Fourier transform of the arithmetic surface measure associated to the Waring--Goldbach problem and provide several applications, including bounds for discrete spherical maximal functions along the primes and distribution results such as ergodic theorems.

关于两个平方、四个四次方、一个五次方与一个七次方的华林-哥德巴赫问题

On Waring-Goldbach problem for two squares, three biquadratics, one quintic and one septic

本文针对华林-哥德巴赫问题，证明每个足够大的奇数可表为两个平方、三个四次方、一个五次方与一个七次方的素数之和，该结果改进了Andrade的相关结论。

On the Waring-Goldbach Problem for One Square and Five Cubes

Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, it is proved that for every sufficiently large even integer $N$, the equation \begin{equation*} N=x^2+p_1^3+p_2^3+p_3^3+p_4^3+p_5^3 \end{equation*} is solvable with $x$ being an almost-prime $\mathcal{P}_6$ and the other variables primes. This result constitutes an improvement upon that of Cai, who obtained the same conclusion, but with $\mathcal{P}_{36}$ in place of $\mathcal{P}_6$.

On the Waring-Goldbach problem with almost equal summands

We use transference principle to show that whenever $s$ is suitably large depending on $k \geq 2$, every sufficiently large natural number $n$ satisfying some congruence conditions can be written in the form $n = p_1^k + \dots + p_s^k$, where $p_1, \dots, p_s \in [x-x^θ, x + x^θ]$ are primes, $x = (n/s)^{1/k}$ and $θ= 0.525 + ε$. We also improve known results for $θ$ when $k \geq 2$ and $s \geq k^2 + k + 1$. For example when $k \geq 4$ and $s \geq k^2 + k + 1$ we have $θ= 0.55 + ε$. All previously known results on the problem had $θ> 3/4$.

Waring-Goldbach Problem: One Square, Four Cubes and Higher Powers

Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, it is proved that, for $12\leqslant b\leqslant 35$ and for every sufficiently large odd integer $N$, the equation \begin{equation*} N=x^2+p_1^3+p_2^3+p_3^3+p_4^3+p_5^4+p_6^b \end{equation*} is solvable with $x$ being an almost-prime $\mathcal{P}_{r(b)}$ and the other variables primes, where $r(b)$ is defined in the Theorem. This result constitutes an improvement upon that of Lü and Mu.

On Waring-Goldbach Problem for Squares, Cubes and Higher Powers

Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, we generalize the result of Vaughan for ternary admissible exponent. Moreover, we use the refined admissible exponent to prove that, for $3\leqslant k\leqslant 14$ and for every sufficiently large even integer $n$, the following equation \begin{equation*} n=x^2+p_1^2+p_2^3+p_3^3+p_4^3+p_5^k \end{equation*} is solvable with $x$ being an almost-prime $\mathcal{P}_{r(k)}$ and the other variables primes, where $r(k)$ is defined in Theorem. This result constitutes a deepening…

Roth--Waring--Goldbach

We use Green's transference principle to show that any subset of the $d$th powers of primes with positive relative density contains nontrivial solutions to a translation-invariant linear equation in $d^2+1$ or more variables, with explicit quantitative bounds.

Waring-Goldbach problem in short intervals

Let $k\geq2$ and $s$ be positive integers. Let $θ\in(0,1)$ be a real number. In this paper, we establish that if $s>k(k+1)$ and $θ>0.55$, then every sufficiently large natural number $n$, subjects to certain congruence conditions, can be written as $$ n=p_1^k+\cdots+p_s^k, $$ where $p_i(1\leq i\leq s)$ are primes in the interval $((\frac{n}{s})^{\frac{1}{k}}-n^{\fracθ{k}},(\frac{n}{s})^{\frac{1}{k}}+n^{\fracθ{k}}]$. The second result of this paper is to show that if $s>\frac{k(k+1)}{2}$ and $θ>0.55$, then almost all integers $n$, subject to certain congruence conditions, have above representat…

关于带有皮亚特谢茨基-沙皮罗集中的素数的二次瓦尔丁-戈尔布问题

On the quadratic Waring-Goldbach problem with primes in Piatetski-Shapiro sets

本文证明了在皮亚特谢茨基-沙皮罗集中的素数条件下，每个足够大的整数n≡5 mod 24都可以表示为五个素数平方的和，改进了之前的结果。

广义黎曼猜想下短区间中的华林-哥德巴赫问题

The Waring-Goldbach problem in short intervals under the Generalized Riemann Hypothesis

在广义黎曼猜想下，研究大整数表示为几乎相等的素数k次幂之和的短区间华林-哥德巴赫问题，证明了分布尺度指数可大于1/2，并给出解数渐近公式，基本解决了该问题。

Linnik's approximation to Goldbach's conjecture, and other problems

We examine the problem of writing every sufficiently large even number as the sum of two primes and at most $K$ powers of 2. We outline an approach that only just falls short of improving the current bounds on $K$. Finally, we improve the estimates in other Waring--Goldbach problems.

具有指定表示函数的华林与华林-哥德包子基

Waring and Waring-Goldbach subbases with prescribed representation function

该研究拓展华林与华林-哥德巴赫问题相关工作，构造出满足指定表示函数渐近或同阶增长性质的$k$次幂集、素数$k$次幂集的子基，还得到了素数幂薄子基的存在性结果。

Four squares of primes and powers of 2

By developing the method of Wooley on the quadratic Waring-Goldbach problem, we prove that all sufficiently large even integers can be expressed as a sum of four squares of primes and 46 powers of 2.

将整数表示为素数混合次幂之和

Representing integers as sums of mixed powers of primes

该研究证明了每个足够大的奇整数可以表示为素数的混合次幂之和，其中包含平方、立方和更高次幂的组合。

关于与Waring问题相关的均值不等式的一个注记

A remark on the mean value inequalities related to Waring's problem

本文在Weyl和的逐点估计假设下，证明了与指数和相关的均值尖锐估计，推进了Waring问题相关均值不等式的研究。

关于自然数加法表示的若干注记

Remarks on additive representations of natural numbers

本文研究互素无平方因子整数下形如 $a p+b P_{2}$ 的自然数表示，给出表示个数的新下界，并推广和改进了短区间、算术级数及哥德巴赫型上界等特殊情形中的已有结果。