Applications of some exponential sums on prime powers: a survey

素数上Kloosterman和的平均界

Average Bounds for Kloosterman Sums Over Primes

证明了素数上Kloosterman分数和的平均值的两个估计。其中第一个改进了Fouvry-Shparlinski和Baker以前的结果。

Asymptotics for sums of functions of primes

This work gives a general approach to the determination of the asymptotic behavior of the sums of functions of primes based on the distribution of primes. It refines the estimate of the remainder term of the asymptotic expansion of the sums of functions of primes. Also, the necessary and sufficient conditions for the existence of these asymptotics are proved in the paper.

素数和的逼近

Approximation of Sums of Primes

在这项工作中，我们考虑了收敛速度非常慢的素数的总和。我们设置为基础，解析素数定理的重新表述，我们使用的黎曼ζ函数的近似值。我们还给出了这些近似的截断误差

A Bombieri -- Vinogradov type result for exponential sums over Piatetski-Shapiro primes

In this paper, we establish a theorem of Bombieri -- Vinogradov type for exponential sums over Piatetski-Shapiro primes $p= [n^{1/γ}]$ with $\frac{865}{886}<γ< 1$.

扭Kloosterman和模素幂的分布

Distribution of twisted Kloosterman sums modulo prime powers

本文研究了模素幂乘特征标扭曲的Kloosterman和。我们表明，由一个初等计算，这些款项成为equidistributed在真正的线相对于一个适当的措施。

Exponential sums over Piatetski-Shapiro primes in arithmetic progressions

Let $γ<1<c$ and $19(c-1)+171(1-γ)<9$. In this paper, we establish an asymptotic formula for exponential sums over Piatetski-Shapiro primes $p=[n^{1/γ}]$ in arithmetic progressions.

与素数III相关的divisor和的较高相关性：k-相关性

Higher correlations of divisor sums related to primes III: k-correlations

我们得到了与素数有关的短因子和的一般k-相关性。

短区间内素数和与素数平方和

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。

Complete Congruences of Jacobi sums of order 2l^2 with prime l

The congruences for Jacobi sums of some lower orders has been treated by many authors in the literature. In this paper we establish the congruences for Jacobi sums of order 2l^2 with odd prime l. These congruences are useful to obtain algebraic and arithmetic characterizations for Jacobi sums of order 2l^2 .

Asymptotics of sums of functions of primes located on an arithmetic progression

We investigate the problem of the distribution of sums of functions of prime numbers located on an arithmetic progression. This problem is closely related to the problem of the distribution of prime numbers on an arithmetic progression. Based on this distribution, a general formula was obtained for the asymptotic estimate of the sums of functions of primes, and also asymptotics was found for the sums of various functions of primes on a geometric progression. Several assertions about asymptotics of sums of functions of prime numbers on a geometric progression are proved. Necessary and sufficien…

Beatty序列中整数的素数和表示

Representations of integers as sums of primes from a Beatty sequence

研究了由固定的Beatty序列$B_{α,β}$表示整数为素数和的问题，其中$α>1$是无理的有限型.

A Brun-Titchmarsh inequality for weighted sums over prime numbers

We prove explicit upper bounds for weighted sums over prime numbers in arithmetic progressions with slowly varying weight functions. The results generalize the well-known Brun-Titchmarsh inequality.

短区间素数与素数上指数和的显式关系

Explicit relations between primes in short intervals and exponential sums over primes

在黎曼假设（RH）的假设下，证明了素数上指数和的截断均方平均渐近公式中的假设误差项与短区间内素数的均方误差项之间的定量关系.我们还注意到，这样的关系与更精确的形式蒙哥马利的配对相关猜想。

Finiteness theorems for universal sums of squares of almost primes

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.

关于两个素数与六个斐波那契数的和

On sums of two primes and six Fibonacci numbers

该研究运用圆法证明，足够大的正整数可表示为两个素数与六个斐波那契数之和、两个素数与七个卢卡斯数之和，还可表示为两个素数、三个斐波那契数与三个卢卡斯数之和。

Signed harmonic sums of integers with $k$ distinct prime factors

We give some theoretical and computational results on "random" harmonic sums with prime numbers, and more generally, for integers with a fixed number of prime factors.

Exponential sums in prime fields for modular forms

The main objective of this article is to study the exponential sums associated to Fourier coefficients of modular forms supported at numbers having a fixed set of prime factors. This is achieved by establishing an improvement on Shparlinski's bound for exponential sums attached to certain recurrence sequences over finite fields.

与素数有关的divisor和的高相关性I：三重相关

Higher correlations of divisor sums related to primes I: triple correlations

我们得到了与素数有关的截断因子和的三重相关。我们还得到了这个因子和在素数上求和时的混合相关性，并给出了它在短区间素数上的一些应用。

Weighted sum formula for multiple harmonic sums modulo primes

In this paper we prove a weighted sum formula for multiple harmonic sums modulo primes, thereby proving a weighted sum formula for finite multiple zeta values. Our proof utilizes difference equations for the generating series of multiple harmonic sums. We also conjecture several weighted sum formulas of similar flavor for finite multiple zeta values.

四次高斯和与元裂隙θ函数

Quartic Gauss sums over primes and metaplectic theta functions

本文改进了四次高斯和的估计，并引入了新的数学工具，同时推测了相关矩的渐进行为。

Character sums over products of prime polynomials

We study sums of Dirichlet characters over polynomials in $\mathbb{F}_q[t]$ with a prescribed number of irreducible factors. Our main results are explicit formulae for these sums in terms of zeros of Dirichlet L-functions. We also exhibit new phenomena concerning Chebyshev-type biases of such sums when the number of irreducible factors is very large.

素数模二次Dirichlet特征和矩的界

Bounds for moments of quadratic Dirichlet character sums of prime moduli

本文在广义黎曼假设下，先对模8p的二次Dirichlet L-函数平移矩建立精确上界，再据此证明素数模二次Dirichlet特征和矩的界。

Prime Sums

We study the properties of certain graphs involving the sums of primes. Their structure largely turns out to relate to the distribution of prime gaps and can be roughly seen in Cramér's model as well. We also discuss generalizations to the Gaussian integers.

素数幂模下二元二次型的特征和

Character sums to prime power moduli evaluated at binary quadratic forms

研究素数幂模下二元二次型的特征和估计，采用p进分析方法，补充Heath-Brown对平方自由模的估计。

András Sárközy未解决问题集中关于剩余类和与素数的问题66与67

Problems 66 and 67 on sums of residue classes and primes of András Sárközy's collection of unsolved problems

本文讨论András Sárközy 2001年未解决问题集中的第66、67题，证明了素数稀疏子集和集的最优结果。

Sums of one prime power and two squares of primes in short intervals

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.

Sums of Kloosterman sums over primes in an arithmetic progression

For $q$ prime, $X \geq 1$ and coprime $u,v \in \mathbb{N}$ we estimate the sums \begin{equation*} \sum_{\substack{p \leq X \substack p \equiv u \hspace{-0.25cm} \mod{v} p \text{ prime}}} \text{Kl}_2(p;q), \end{equation*} where $\text{Kl}_2(p;q)$ denotes a normalised Kloosterman sum with modulus $q$. This is a sparse analogue of a recent theorem due to Blomer, Fouvry, Kowalski, Michel and Milićević showing cancellation amongst sums of Kloosterman sums over primes in short intervals. We use an optimisation argument inspired by Fouvry, Kowalski and Michel. Our argument compares three different bo…

关于Chen素数子集的和

On sums of subsets of Chen primes

本文证明了：若$A$是具有正相对密度$α$的Chen素数子集，则$A+A$在自然数中至少有$cαe^{-c^\prime\log(1/α)^{2/3}(\log\log(1/α))^{1/3}}$的正上密度.

Distribution of short sums of classical Kloosterman sums of prime powers moduli

Corentin Perret-Gentil proved, under some very general conditions, that short sums of $\ell$-adic trace functions over finite fields of varying center converges in law to a Gaussian random variable or vector. The main inputs are P.~Deligne's equidistribution theorem, N.~Katz' works and the results surveyed in \cite{MR3338119}. In particular, this applies to $2$-dimensional Kloosterman sums $\mathsf{Kl}_{2,\mathbb{F}_q}$ studied by N.~Katz in \cite{MR955052} and in \cite{MR1081536} when the field $\mathbb{F}_q$ gets large. \par This article considers the case of short sums of normalized classic…

四类几乎质数长度的广义多边形数之和

Sums of four generalized polygonal numbers of almost prime length

研究四类受限质数因子数的广义多边形数之和，证明大数可表示为此类和，参数限制最多988个质因子。

素数幂模的矩阵 Kloosterman 和

Matrix Kloosterman sums modulo prime powers

本文研究素数幂模的矩阵 Kloosterman 和，将既有素数模结果推广到素数幂情形，给出最优界，并指出其与 $\mathrm{GL}_n$ 上 horocyclic 流的联系。

素幂模下的Kloosterman和的乘积和

Sums of products of Kloosterman sums to prime power moduli

本文针对素幂模下的Kloosterman和乘积和，证明了新的强幂节省界，通过利用p进可微性的初始项估计完全指数和的方法实现，为相关数论问题提供了量化的对齐分析。

Salié和的移位双线性和与移位素数的模平方根分布

Shifted bilinear sums of Salié sums and the distribution of modular square roots of shifted primes

研究Salié和的移位双线性和的上界，并分析移位素数的模平方根分布，揭示$b\not \equiv 0 \bmod q$时的新困难。

关于短区间中素数指数和的一个条件性结果

A conditional result on exponential sums over primes in short intervals

本文在广义 Riemann 假设及 Dirichlet L 函数密度假设同时成立的条件下，给出短区间中素数指数和的一个条件性结果。

多素数和

Sums of many primes

在广义黎曼假设成立的条件下，证明了一个关于整数表示为$k\geq 5$素数之和的个数的显式公式。我们在这样的公式中的误差项通过一些对数因子改进了Friedlander-Goldston的类似结果。

关于稀疏素子集的和

On sums of sparse prime subsets

对于任意的$c_0>0$，如果$A$是小于$x$的素数的子集，其基数为$δx (\log x)^{-1}$和$δ\geq (\log x)^{-c_0}$，则存在正的常数$c$，使得$A+A$的基数大于$c\, δx (\log\log x)^{-1}$。

Binomial Character Sums Modulo Prime Powers

We show that the binomial and related multiplicative character sums $$ \sum_{\stackrel{x=1}{(x,p)=1}}^{p^m} χ(x^l(Ax^k +B)^w),\hspace{3ex} \sum_{x=1}^{p^m} χ_1 (x)χ_2(Ax^k +B), $$ have a simple evaluation for large enough $m$ (for $m\geq 2$ if $p\nmid ABk$).

移位素数上的Character和

Character sums over shifted primes

对于整数$q$，设$χ$是本原乘法特征标$\pmod q.$。对于与$q$互质的整数$a$，我们得到了和$$\sum_{n\le N}Λ(n)χ(n+a),$$的一个新的界，其中$Λ(n)$是von Mangoldt函数。这个界改进和推广了Friedlander，Gong和Shparlinski的一个结果

Primes from sums of two squares and missing digits

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.

关于素数II的平方和

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。

素数高阶和的密度定理

A Density Theorem for Higher Order Sums of Prime Numbers

该研究证明下密度严格大于1/2的素数子集可表所有充分大偶数为其4个素数之和，还推广到k≥4个加项及k≥4个不同素数子集的情形，以初等组合引理为核心新贡献，延伸了此前的两、三素数和相关研究。

Sign changes of Kloosterman sums with almost prime moduli

We prove that the Kloosterman sum $S(1,1;c)$ can change sign infinitely often as $c$ runs over squarefree moduli with at most 10 prime factors, which improves the previous results of E. Fouvry and Ph. Michel, J. Sivak-Fischler and K. Matomäki, replacing 10 by 23, 18 and 15, respectively. The method combines the Selberg sieve, equidistribution of Kloosterman sums and spectral theory of automorphic forms.

探索大正整数表示为素数幂与整数幂之和：基于正密度子集的分析

Exploring the Representation of Large Positive Integers as Sums of Prime Powers and Integer Powers: Analysis with Positive Density Subsets

本文利用转换原理研究大正整数表示为素数幂与整数幂之和的问题，其中素数和整数幂分别取自正密度子集，拓展了加法数论中的表示理论。

关于素数的不同次幂与2的幂之和

On sums of unequal powers of primes and powers of 2

本文证明了充分大的偶数可表示为素数平方、立方、四次方及2的幂之和，且至少5.313%的奇数可表示为素数平方、立方与四次方之和，改进了R. Zhang的结果。

Incomplete Gauss sums modulo primes

We obtain a new bound for incomplete Gauss sums modulo primes. Our argument falls under the framework of Vinogradov's method which we use to reduce the problem under consideration to bounding the number of solutions to two distinct systems of congruences. The first is related to Vinogradov's mean value theorem, although the second does not appear to have been considered before. Our bound improves on current results in the range $N\ge q^{2k^{-1/2}+O(k^{-3/2})}$.

关于素数加平方或立方之和的计算结果

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。

关于素数上的某些求和与里斯函数

On certain sums over primes and the Riesz function

本文通过Gram级数导数与里斯函数的关系，探讨素数求和近似中的Mobius函数级数，推导出一般积分关系。

实指数和与素数间隔

Real exponential sums over primes and prime gaps

研究证明在短区间内存在素数，解决了长期存在的问题，并验证了关于素数分布的旧猜想。

A conjecture of Erdos, supersingular primes and short character sums

If $k$ is a sufficiently large positive integer, we show that the Diophantine equation $$n (n+d) \cdots (n+ (k-1)d) = y^{\ell}$$ has at most finitely many solutions in positive integers $n, d, y$ and $\ell$, with $\operatorname{gcd}(n,d)=1$ and $\ell \geq 2$. Our proof relies upon Frey-Hellegouarch curves and results on supersingular primes for elliptic curves without complex multiplication, derived from upper bounds for short character sums and sieves, analytic and combinatorial.

几乎素数与可表示为两平方和加1的素数

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。

二元加法方程的素数解与特征和混合矩

Prime Solutions to a Binary Additive Equation and Mixed Moments of Character Sums

本文通过泊松求和与Kloosterman分数和均值估计，得到二元加法方程素数解的渐近公式，并应用于特征和混合四阶矩，获得非平凡界。

Sums of four squares of primes

Let $E(N)$ denote the number of positive integers $n \le N$, with $n \equiv 4 \pmod{24}$, which cannot be represented as the sum of four squares of primes. We establish that $E(N)\ll N^{11/32}$, thus improving on an earlier result of Harman and the first author, where the exponent $7/20$ appears in place of $11/32$.

Counting primes by sums of frequencies

We introduce the sequence $(a_n) \subset (0,1]$ and prove that the asymptotic behaviour of $\sum_{k=1}^n a_k$ is the same than $π(n)$, the prime-counting function. We also obtain that $π(n) \sim n a_n$ and we estimate $\frac{1}{a_n}-\frac{n}{π(n)}$ showing that $\lim_{n \rightarrow \infty} \frac{1}{a_n}-\frac{n}{π(n)}$ is convergent.

Universal elliptic Gauß sums for Atkin primes in Schoof's algorithm

This work builds on earlier results. We define universal elliptic Gauß sums for Atkin primes in Schoof's algorithm for counting points on elliptic curves. Subsequently, we show these quantities admit an efficiently computable representation in terms of the $j$-invariant and two other modular functions. We analyse the necessary computations in detail and derive an alternative approach for determining the trace of the Frobenius homomorphism for Atkin primes using these pre-computations. A rough run-time analysis shows, however, that this new method is not competitive with existing ones.