S. I. Dimitrov
Let $[\, \cdot\,]$ be the floor function. In this paper, we show that when $1<c<37/36$, then every sufficiently large positive integer $N$ can be represented in the form \begin{equation*} N=[p^c_1]+[p^c_2]+[p^c_3]\,, \end{equation*} where $p_1,p_2,p_3$ are primes close to squares.
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Large gaps between consecutive prime numbers containing square-free numbers and perfect powers of prime numbers
我们证明了K. Ford，D. R. Heath-Brown和S. Konyagin关于无平方数的素数回避和素数的完全幂。
We prove explicit bounds for the number of sums of consecutive prime squares below a given magnitude.
Sums of primes and squares of primes in short intervals
令$\mathbf H_2$表示偶数集合$n \not\equiv 1 \pmod 3$。我们证明了当$H \ge X^{0.33}$，几乎所有的整数$n \in \mathbf H_2$，$X < n \le X + H$都可以表示为一个素数与一个素数的平方之和.我们也证明了一个类似的结果，三个平方和的素数。
Sum of one prime and two squares of primes in short intervals
在黎曼假设的假设下，证明了区间$[N, N + H]$包含一个整数，它是一个素数和两个素数的平方之和，条件是$H \ge C (\log N)^{4}$，其中$C > 0$是有效常数.
Squarefree smooth numbers and Euclidean prime generators
我们表明，对于每个素数p > 7，每一个剩余模p可以表示一个squarefree数最大的素因子在最多p。我们给出了两个应用程序递归素生成类似于一个欧几里德用来证明素数的无限。
On the Sum of a Prime and a Square-free Number
证明了任何大于2的整数都可以写成一个素数和一个无平方数之和。
Difference of powers of consecutive primes which are perfect squares
研究了连续素数$p$和$q$（$p > q$），对它们存在一对自然数$(x,y)$使得$p^x-q^y$是完满平方，并作了一些证明.
Let $[\, \cdot\,]$ be the floor function. In this paper we show that when $1<c<\frac{3849}{3334}$, then there exist infinitely many prime numbers of the form $[n^c]$, where $n$ is square-free.
Four squares of primes and powers of 2
通过发展Wooley关于二次Waring-哥德巴赫问题的方法，我们证明了所有充分大的偶数整数都可以表示为4个素数的平方与2的46次幂之和.
On the sum of a prime and a square-free number co-prime to any integer with at most two prime factors
研究证明大于2的自然数可表示为素数与特定平方数之和，并明确给出了渐进行为结果。
In this paper we study quadratic forms which are universal when restricted to almost prime inputs, establishing finiteness theorems akin to the Conway--Schneeberger 15 theorem.
We show that every sufficiently large integer is a sum of a prime and two almost prime squares, and also a sum of a smooth number and two almost prime squares. The number of such representations is of the expected order of magnitude. We likewise treat representations of shifted primes p-1 as sums of two almost prime squares. The methods involve a combination of analytic, automorphic and algebraic arguments to handle representations by restricted binary quadratic forms with a high degree of uniformity.
Large Sieve Inequalities for Special Characters to Prime Square Moduli
本文对模为素数平方的一些特殊特征标建立了一个大筛法不等式。我们的结果给出了在一定范围内的非平凡估计。
Every odd prime number p can be written in exactly (p + 1)/2 ways as a sum ab+cd of two ordered products ab and cd such that min(a, b) > max(c, d). An easy corollary is a proof of Fermat's Theorem expressing primes in 1 + 4N as sums of two squares.
We prove that every positive integer $n$ which is not equal to $1$, $2$, $3$, $6$, $11$, $30$, $155$, or $247$ can be represented as a sum of a squarefree number and a prime not exceeding $\sqrt{n}$.
We obtain an asymptotic formula for the number of ways to represent every reduced residue class as a product of a prime and square-free integer. This may be considered as a relaxed version of a conjecture of Erdös, Odlyzko, and Sárközy.
Let $k\ge 1$ be an integer. We prove that a suitable asymptotic formula for the average number of representations of integers $n=p_{1}^{k}+p_{2}^{2}+p_{3}^{2}$, where $p_1,p_2,p_3$ are prime numbers, holds in intervals shorter than the ones previously known.
Digital functions along Squares of Prime Numbers
本文将Gelfond猜想中沿素数与沿平方的分布问题相结合，证明沿平方的q-可乘函数素数定理，并由此得到沿素数平方的良好分布结果。
On the Sum of the Square of a Prime and a Square-Free Number
我们证明了任何整数$n \geq 10$使得$n \not\equiv 1 \text{mod} 4$可以写成素数和无平方数的平方和。这使得明确的一个定理Erdos的每一个足够大的整数这种类型可以写在这样一种方式。我们的证明要求我们为等差数列中的素数构造新的显式结果。因此，我们使用第二作者关于GRH的数值计算来扩展Ramaré-Rumely的显式界。
Square-free values of polynomials evaluated at primes over a function field
我们研究了一个经典问题的函数域版本，该问题涉及多项式在素数处的无平方值。我们证明了对于无平方多项式$f\in \mathbb{F}_q[t][x]$，存在一个极限密度为素数$P \in \mathbb{F}_q[t]$的次数$n$的$n\to \infty$，使得$f(P)$是无平方的.在整数上，只有当$f$的所有不可约因子的次数最多为3时，才知道类似的结果。
On primes amongst squares of naturals
分析了不等式$\sqrt{P_{k+1}}-\sqrt{P_{k}}<1,\ k\in\mathbb{N}$，讨论了素数在任意区间$(r,s),\ r<s,\ r,s\in\mathbb{R}$上的存在性，最后讨论了自然数平方之间的素数问题.
The exceptional set of a prime and a square
本文证明了素数加平方数表示中例外集的上界为$O(X^{4/5})$，改进了相关估计。
We study the problem of writing Gaussian primes as the sum of two squares, both of which are interesting arithmetically, in particular, when one is the square of a prime and the other the square of an almost-prime.
Almost primes and primes that are sums of two squares plus 1
该研究用向量筛框架（含半线性筛、线性筛）得出满足p-1为两平方和且p+2素因子个数有界的素数p≤x的下界。
Squares in arithmetic progressions and infinitely many primes
我们给出了一个新的证明，有无穷多个素数，依赖于范德瓦尔登定理着色的整数，和费马大定理，不能有四个平方的等差数列。我们继续讨论这些想法在过去的其他地方聚集在一起。
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。
Let $c$ be fixed with $1<c<35/34$. In this paper we prove that for every sufficiently large real number $N$ and a small constant $\varepsilon>0$, the diophantine inequality \begin{equation*} |p_1^c+p_2^c+p_3^c-N|<\varepsilon \end{equation*} is solvable in primes $p_1,\,p_2,\,p_3$ near to squares.
Explicit unobstructed primes for modular deformation problems of squarefree level
设f是数域K中权至少为2且具有傅里叶系数的无平方水平的新形式。我们给出明确的界限，根据与其他newforms的f的同余，在一组素数λ K的变形问题与模λ伽罗瓦表示的f是阻碍。我们包括一些明确的例子。
Let $p$ be a prime. We define $S(p)$ the smallest number $k$ such that every positive integer is a sum of at most $k$ squares of integers that are not divisible by $p$. In this article, we prove that $S(2)=10$, $S(3)=6$, $S(5)=5$, and $S(p)=4$ for any prime $p$ greater than $5$. In particular, it is proved that every positive integer is a sum of at most four squares not divisible by $5$, except the unique positive integer $79$.
Computational results on sums of a prime with squares or cubes
本文通过计算验证了素数加两个平方数或立方数的表示猜想，证明了特定范围内所有整数均可表示，并利用椭圆曲线整点方法实现。
Let $\mathcal{A}'$ be the set of integers missing any three fixed digits from their decimal expansion. We produce primes in a thin sequence by proving an asymptotic formula for counting primes of the form $p = m^2 + \ell^2$, with $\ell \in \mathcal{A}'$. The proof draws on ideas from the work of Friedlander-Iwaniec on primes of the form $p = x^2+y^4$, as well as ideas from the work of Maynard on primes with restricted digits.
A density theorem for prime squares
该论文证明在素数集合具有足够大相对下密度时，每个充分大的同余类整数可表示为该集合中固定个数素数平方之和。
On the representation of integer as sum of a square-free number and a prime of special type
本文证明存在无穷多个整数可表示为一个无平方因子整数与一个满足特定丢番图不等式条件的素数之和，其中系数α为无理数。
Sums of four squares of primes
令$E(N)$表示正整数$n \le N$的个数，其中$n \equiv 4 \pmod{24}$不能表示为四个素数的平方和。我们建立了$E(N)\ll N^{11/32}$，从而改进了Harman和第一作者的早期结果，其中指数$7/20$代替了$11/32$。
The prime geodesic theorem in square mean
本文加强了关于$\mathrm{PSL}_2(\mathbb{Z})$素测地定理误差项平方均值的研究结果，并进一步发展了该结果的短区间版本。
We prove upper bounds for the mean square of the remainder in the prime geodesic theorem, for every cofinite Fuchsian group, which improve on average on the best known pointwise bounds. The proof relies on the Selberg trace formula. For the modular group we prove a refined upper bound by using the Kuznetsov trace formula.
A Diophantine problem with a prime and three squares of primes
证明了：若$λ_1$，$λ_2$，$λ_3$和$λ_4$是非零实数，且符号不相同，$λ_1 / λ_2$是无理数，$\varpi$是任意实数，则对任意$\eps > 0$，不等式$ \bigl|λ_1 p_1 + λ_2 p_2^2 + λ_3 p_3^2 + λ_4 p_4^2 + \varpi \bigr| \le \bigl(\max_j p_j \bigr)^{-1 / 18 + \eps} $在素变量$p_1$，.中有无穷多个解，$p_4$
We show that there are infinitely many primes $p$ such that $p-1$ is divisible by a square $d^2 \geq p^θ$ for $θ=1/2+1/2000.$ This improves the work of Matomäki (2009) who obtained the result for $θ=1/2-\varepsilon$ (with the added constraint that $d$ is also a prime), which improved the result of Baier and Zhao (2006) with $θ=4/9-\varepsilon.$ Similarly as in the work of Matomäki, we apply Harman's sieve method to detect primes $p \equiv 1 \, (d^2)$. To break the $θ=1/2$ barrier we prove a new bilinear equidistribution estimate modulo smooth square moduli $d^2$ by using a similar argument as …
We use recent bounds on bilinear sums with modular square roots to study the distribution of solutions to congruences $x^2 \equiv p \pmod q$ with primes $p\le P$ and integers $q \le Q$. This can be considered as a combined scenario of Duke, Friedlander and Iwaniec with averaging only over the modulus $q$ and of Dunn, Kerr, Shparlinski and Zaharescu with averaging only over $p$.
On the mean square gap between primes
该数学论文改进了连续素数差平方平均值的上界，采用Harman筛法等工具，将Peck的O(x^{0.25+ε})界优化为O(x^{0.23+ε})。
On monochromatic representation of sums of squares of primes
当素数的平方序列用$K$颜色着色时，其中$K \geq 1$是整数，设$s(K)$是最小的整数，使得每个足够大的整数可以写成不超过$s(K)$的素数平方的和，所有的素数都是相同的颜色。我们表明，$s(K) \ll K \exp\left(\frac{(3\log 2 + {\rm o}(1))\log K}{\log \log K}\right)$为$K \geq 2$。这改进了$s(K) \ll_ε K^{2 +ε}$，它是$s(K)$的最佳可用上界。
Primes in the intervals between primes squared
连续素数平方之间的短间隔集具有令人愉快的--但似乎未被利用的--性质，即每个间隔$s_k:=\{p_k^2, \dots,p_{k+1}^2-1\}$都被$k$第一个素数完全筛选。在这里，我们利用这一基本特征，并提出证据的猜想，即$π_k \sim |s_k|/ \log p_{k+1}^2$，其中$π_k$是素数在$s_k$的数量;或更严格的，即$y=x^{1/2}$是必要的和充分的素数定理是有效的长度为$y$的间隔。此外，我们提出并证实，素数计数函数$π(x)$是最好的理解为相关的随机变量$π_k$的总和。在此假设下，我们推导出$π(p_{k+1}^2)=\sum_{j=1}^k π_j$的理论方差，由此我们推测$|π({x})-\textrm{li}(x)| =O(\sqrt{\textrm{li}(x)})$。从我们的研究中出现的观点是，连续素数之间的间隔平方持有进一步理解素数分布的关键;如所证明的，这种观点也建立了有利于黎曼假设的强有力的支持。
On a Diophantine problem with one prime, two squares of primes and $s$ powers of two
本文改进了W.P. Li和Wang关于$ λ_1p_1 + λ_2p_2^{2} + λ_3p_3^{2} + μ_1 2^{m_1} +...+ μ_s 2^{m_s}, $形式的值的一个结果，其中$p_1,p_2,p_3$是素数，$m_1,..., m_s$是正整数，$λ_1,λ_2,λ_{3}$是非零实数，不都是同号的，$λ_2 / λ_3$是无理数，而$λ_i/μ_i \in \Q$，对于$i\in\{1,2,3\}$ .
Let $\ell \geq 5$ be a prime and let $N$ be a non-squarefree integer not divisible by $\ell$. For a rational Eisenstein prime $\mathfrak{m}$ of the Hecke ring $\mathbb{T}(N)$ of level $N$ acting on $J_0(N)$, we precisely compute the dimension of the kernel $J_0(N)[\mathfrak{m}]$ under a mild assumption. In the case of level $qr^2$ which violates our mild assumption, we propose a conjecture based on Sage computations. Assuming this conjecture, we complete our computation in all the remaining cases.
Uniformly for small $q$ and $(a,q)=1$, we obtain an estimate for the weighted number of ways a sufficiently large integer can be represented as the sum of a prime congruent to $a$ modulo $q$ and a square-free integer. Our method is based on the notion of local model developed by Ramaré and may be viewed as an abstract circle method.
A remark on the conditional estimate for the sum of a prime and a square
Hardy和Littlewood指出，每一个足够大的整数要么是平方，要么是素数和平方的和。令$E(x)$为不满足该条件的直到$x\ge4$的正整数的数目。在广义黎曼假设下，我们用$A=3/2$证明了$E(x)\ll x^{1/2}(\log x)^A(\log\log x)^4$.这是对Mikawa（1993）和Perelli-Zaccagnini（1995）先前的评论的一个小改进，它们分别声明了$A=4,3$。
In 1951, Linnik proved the existence of a constant $K$ such that every sufficiently large even number is the sum of two primes and at most $K$ powers of 2. Since then, this style of approximation has been considered for problems similar to the Goldbach conjecture. One such problem is the representation of a sufficiently large even number as a sum of four squares of primes and at most $k$ powers of two. In 2014, Zhao proved this to be true with $k = 46$. In this paper, we reduce this to $k = 31$.
Short intervals asymptotic formulae for binary problems with primes and powers, II: density $1$
我们证明，适当的渐近公式在短时间内举行的问题，表示一个整数作为一个素数平方和平方，或一个素数平方。这样的结果得到假设黎曼假设和在无条件的情况下。
A Cesàro average for an additive problem with an arbitrary number of prime powers and squares
该研究针对整数表示为素数幂与平方数之和的加性问题，拓展改进了带Cesàro权加权平均的现有结果，覆盖所有已研究情形并借助所选技术得到最优结论。
A binary additive equation with prime and square-free number
本文证明了在 $1<c<\frac{82}{79}$ 条件下，每个充分大的正整数可表示为素数幂与无平方数幂的取整和。
Clusters of primes with square-free translates
设$\mathcal{R}$是满足适当局部条件的整数的有限集。我们证明了所有$b \in \mathcal{R}$的长簇素数$p$在有界长度区间上的存在性，且$p-b$无平方。此外，我们还可以强制要求聚簇中的素数$p$满足以下条件之一：（1）$p$位于短区间$[N, N+N^{\frac{7}{12}+ε}]$内，（2）$p$属于给定的非齐次Beatty序列，（3）当$c \in (\frac{8}{9},1)$固定时，$p^c$位于长度为$p^{-1+c+ε}$的指定区间mod $1$内。
On the sum of a prime and a square-free number with divisibility conditions
本文研究素数与无平方因子数之和，在施加整除条件时给出显式与渐近结果，并证明对特定范围的偶数 $k$，任何偶数 $n\geq 40$ 均可表示为素数与同 $k$ 互素的无平方因子数之和。
On the Cesàro average of the numbers that can be written as sum of a prime and two squares of primes
令$Λ\left(n\right)$为冯·曼戈尔特函数，$r_{SP}\left(n\right)=\sum_{m_{1}+m_{2}^{2}+m_{3}^{2}=n}Λ\left(m_{1}\right)Λ\left(m_{2}\right)Λ\left(m_{3}\right)$为计数函数，用于可以被写为一个素数和两个素数的平方和的数。令$N$为足够大的整数，令$k>3/2$和令$M_{i}\left(N,k\right),\, i=1,\dots,4$为取决于$Γ(s)$的合适参数。我们证明了$$\sum_{n\leq N}r_{SP}\left(n\right)\frac{\left(N-n\right)^{k}}{Γ\left(k+1\right)}=M_{1}\left(N,k\right)+M_{2}\left(N,k\right)+M_{3}\left(N,k\right)+M_{4}\left(N,k\right)+O\left(N^{k+1}\right).$$
On sums of squares of primes II
在本文中，我们继续我们的研究，开始在第一部分，例外的一组整数，不受限制的基本同余条件，这不能表示为总和的三个或四个平方的素数。我们在第一篇论文中纠正了一个严重的疏忽，但在所需的指数和估计方面取得了进一步的进展，同时对以前采用的筛分技术进行了修饰。这导致我们的边界的最大尺寸的例外集的改进。
In this note, we study the problem of existence of sequences of consecutive 1's in the periodic part of the continued fractions expansions of square roots of primes. We prove unconditionally that, for a given $N\gg 1$, there are at least $N\log^{-3/2}N$ prime numbers $p\leq N$ such that the continued fraction expansion of $\sqrt{p}$ contains three consecutive 1's on the beginning of the periodic part. We also present results of our computations related to the considered problem and some related problems, formulate several open questions and conjectures and get some results under the assumption…
On the Existence of Integers with at Most 3 Prime Factors Between Every Pair of Consecutive Squares
本文通过改进Dudek和Johnston的结果，证明了每对连续平方之间存在一个具有最多3个素因子的整数，从而在素数分布理论中取得进展。
Selberg sieve weights and sign changes of Kloosterman sums. II. Square-free moduli with at most four prime factors
本文证明对每个符号，存在大量（\\(\gg X/\log X\\)）至多四个素因子的无平方因子模\\(q\\)，其归一化Kloosterman和取该符号，方法结合Selberg筛与等分布及均方估计。
Let $η>0$ be a fixed positive number, let $N$ be a sufficiently large number. In this paper, we study the second moment of the sum of Hecke eigenvalues over primes in short intervals (whose length is $η\log N$) on average (with some weights) over the family of weight $k$ holomorphic Hecke cusp forms. We also generalize the above result to Hecke-Maass cusp forms for $SL(2,\mathbb{Z})$ and $SL(3,\mathbb{Z}).$ By applying the Hardy-Littlewood prime 2-tuples conjecture, we calculate the exact values of the mean values.
Asymptotic formula for the sum of a prime and a square-full number in short intervals
本文研究素数与平方满数之和的表示函数，证明其在长度略大于 $X^{1/2}$ 的短区间内求和时满足渐近公式。
On sums of Hecke-Maass eigenvalues squared over primes in short intervals
我们证明了一个统一的估计的总和Hecke-马斯特征值平方素数在短时间内，可以被视为一个模拟Hoheisel的经典素数定理的所有真正的解析尖点形式。我们的论点是仿照我们的治疗林尼克的最小素数定理等差数列（塔塔LN 72），并依赖于最近的作品在理论的自守表示。我们强调，常数在目前的工作，包括那些隐含的，都是普遍的，有效地计算，虽然我们不特别注意数值精度。在第三个版本中，我们对主断言进行了改进，并添加了更多细节。在这个新的版本中，一些扩充有关对称L-函数的对数导数的界限。然而，这只是为了使我们的论点更容易理解。
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