William D. Banks, Ahmet M. Guloglu, C. Wesley Nevans
We study the problem of representing integers as sums of prime numbers from a fixed Beatty sequence $B_{α,β}$, where $α>1$ is irrational and of finite type.
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We study sums with multiplicative functions that take values over a non-homogenous Beatty sequence. We then apply our result in a few special cases to obtain asymptotic formulas such as the number of integers in a Beatty sequence representable as a sum of two squares up to a given magnitude.
We estimate multiplicative character sums taken on the values of a non-homogeneous Beatty sequence $\{[αn + β] : n =1,2,... \}$, where $α,β\in\R$, and $α$ is irrational. Our bounds are nontrivial over the same short intervals for which the classical character sum estimates of Burgess have been established.
In this paper, we study the gaps between primes in Beatty sequences following the methods in the recent breakthrough of J. Maynard.
We prove an upper bound for the least prime in an irrational Beatty sequence. This result may be compared with Linnik's theorem on the least prime in an arithmetic progression.
Let $k \ge 2$ and $α_1, β_1, ..., α_k, β_k$ be reals such that the $α_i$'s are irrational and greater than 1. Suppose further that some ratio $α_i/α_j$ is irrational. We study the representations of an integer $n$ in the form $$ p_1 + p_2 + ... + p_k = n, $$ where $p_i$ is a prime from the Beatty sequence $$ \mathcal B_i = \left\{n \in \mathbb N : n = [ α_i m + β_i ] \text{for some} m \in \mathbb Z \right\}. $$
Let $α>1$ be irrational and of finite type, $β\in\mathbb{R}$. In this paper, it is proved that for $R\geqslant13$ and any fixed $c\in(1,c_R)$, there exist infinitely many primes in the intersection of Beatty sequence $\mathcal{B}_{α,β}$ and $\lfloor n^c\rfloor$, where $c_R$ is an explicit constant depending on $R$ herein, $n$ is a natural number with at most $R$ prime factors, counted with multiplicity.
Let $α,β$ be real numbers such that $α>1$ is irrational and of finite type, and let $c$ be a real number in the range $1<c<\frac{14}{13}$. In this paper, it is shown that there are infinitely many Piatetski-Shapiro primes $p = \left\lfloor n^c \right\rfloor$ in the non-homogenous Beatty sequence $\big(\left\lfloorαm+β\right\rfloor\big)_{m=1}^\infty$.
We study the prime numbers that lie in Beatty sequences of the form $\lfloor αn + β\rfloor$ and have prescribed algebraic splitting conditions. We prove that the density of primes in both a fixed Beatty sequence and a Chebotarev class of some Galois extension is precisely the product of the densities $α^{-1}\cdot\frac{|C|}{|G|}$. Moreover, we show that the primes in the intersection of these sets satisfy a Bombieri--Vinogradov type theorem. This allows us to prove the existence of bounded gaps for such primes. As a final application, we prove a common generalization of the aforementioned bound…
Suppose that $α_1, α_2,β_1, β_2 \in\mathbb{R}$. Let $α_1, α_2 > 1$ be irrational and of finite type such that $1, α_1^{-1}, α_2^{-1}$ are linearly independent over $\mathbb{Q}$. Let $c$ be a real number in the range $1 < c < 12/11$. In this paper, it is proved that there exist infinitely many primes in the intersection of Beatty sequences $\mathcal{B}_{α_1,β_1} = \lfloorα_1 n + β_1\rfloor, \mathcal{B}_{α_2, β_2} = \lfloorα_2 n + β_2\rfloor$ and the Piatetski-Shapiro sequence $\mathscr{N}^{(c)} = \lfloor n^c\rfloor$. Moreover, we also give a sketch proof of Piatetski-Shapiro primes in the inter…
For a polynomial $g(x)$ of deg $k \geq 2$ with integer coefficients and positive integer leading coefficient, we prove an upper bound for the least prime $p$ such that $g(p)$ is in non-homogeneous Beatty sequence $\lbrace \lfloor αn+β\rfloor : n=1,2,3, \dots \rbrace$, where $α, β\in \mathbb{R}$ with $α>1$ is irrational and we prove an asymptotic formula for the number of primes $p$ such that $g(p)=\lfloor αn+β\rfloor.$ Next we obtain an asymptotic formula for number of primes $p$ of the form $p=\lfloor αn+β\rfloor$ which also satisfies $p \equiv f \pmod d$ where $f, d$ are integers with $1\leq…
Fix irrational numbers $α,\hatα>1$ of finite type and real numbers $β,\hatβ\ge 0$, and let $B$ and $\hat B$ be the Beatty sequences $$ B:=(\lfloorαm+β\rfloor)_{m\ge 1}\quad\text{and}\quad\hat B:=(\lfloor\hatαm+\hatβ\rfloor)_{m\ge 1}. $$ In this note, we study the distribution of pairs $(p,p^\sharp)$ of consecutive primes for which $p\in B$ and $p^\sharp\in\hat B$. Under a strong (but widely accepted) form of the Hardy-Littlewood conjectures, we show that $$ \big|\{p\le x:p\in B\text{ and }p^\sharp\in\hat B\}\big|=(α\hatα)^{-1}π(x)+O\big(x(\log x)^{-3/2+ε}\big), $$ where $π(x)$ is the prime cou…
A study of certain Hamiltonian systems has lead Y. Long to conjecture the existence of infinitely many primes of the form $p=2[αn]+1$, where $1<α<2$ is a fixed irrational number. An argument of P. Ribenboim coupled with classical results about the distribution of fractional parts of irrational multiples of primes in an arithmetic progression immediately imply that this conjecture holds in a much more precise asymptotic form. Motivated by this observation, we give an asymptotic formula for the number of primes $p=q[αn+β]+a$ with $n\le N$, where $α,β$ are real numbers such that $α$ is positive a…
Integer sequences of the form $\lfloor n^c\rfloor$, where $1<c<2$, can be locally approximated by sequences of the form $\lfloor nα+β\rfloor$ in a very good way. Following this approach, we are led to an estimate of the difference \[\sum_{n\leq x}φ\left(\lfloor n^c\rfloor\right)-\frac 1c\sum_{n\leq x^c}φ(n)n^{\frac 1c-1},\] which measures the deviation of the mean value of $φ$ on the subsequence $\lfloor n^c\rfloor$ from the expected value, by an expression involving exponential sums. As an application we prove that for $1<c\leq 1.42$ the subsequence of the Thue-Morse sequence indexed by $\lfl…
Prime Avoidance Property of $k$-th Powers of Piatetski-Shapiro Primes
本文研究Piatetski-Shapiro素数$k$次幂的素数避让性质，扩展了作者先前关于素数$k$次幂及Beatty序列中素数的同类结论。
Modular hyperbolas and Beatty sequences
本文研究素数模下落在 Beatty 序列中的模双曲线点，借助 Banks–Shparlinski 与改造的 Kloosterman 方法，建立坐标最大值及相关指数和序列周期自相关的界。
On primes in special sequences with applications to Carmichael numbers
本文利用等差数列中与Λ(n)相关的指数和，改进了非齐次Beatty序列和Piatetski-Shapiro序列上素数分布的结果，并将其应用于改进卡迈克尔数相关结论。
We generalize results of Alladi, Dawsey, and Sweeting and Woo for Chebotarev densities to general densities of sets of primes. We show that if $K$ is a number field and $S$ is any set of prime ideals with natural density $δ(S)$ within the primes, then \[ -\lim_{X \to \infty}\sum_{\substack{2 \le \operatorname{N}(\mathfrak{a})\le X\\ \mathfrak{a} \in D(K,S)}}\frac{μ(\mathfrak{a})}{\operatorname{N}(\mathfrak{a})} = δ(S), \] where $μ(\mathfrak{a})$ is the generalized Möbius function and $D(K,S)$ is the set of integral ideals $ \mathfrak{a} \subseteq \mathcal{O}_K$ with unique prime divisor of min…
The main results extend to sums over primes in a short interval earlier estimates by the author for "long" Weyl sums over primes.
We obtain a nontrivial bound for cancellations between the Kloosterman sums modulo a large prime power with a prime argument running over very short interval, which in turn is based on a new estimate on bilinear sums of Kloosterman sums. These results are analogues of those obtained by various authors for Kloosterman sums modulo a prime. However the underlying technique is different and allows us to obtain nontrivial results starting from much shorter ranges.
We prove explicit bounds for the number of sums of consecutive prime squares below a given magnitude.
Barban-Davenport-Halberstam type theorems for exponential sums and Piatetski-Shapiro primes
该研究在数论领域建立了三类Barban-Davenport-Halberstam型定理，分别适用于素数指数和、Piatetski-Shapiro素数及该类素数上的指数和，拓展了相关定理的应用范围。
We obtain new results on additive properties of the set $$ I^{-1}= \{x^{-1}: \quad x\in I\} $$ where $I$ is an arbitrary interval in the field of residue classes modulo a large prime $p$. We combine our results with multilinear exponential sum estimates and obtain new results on incomplete multilinear Kloosterman sums.
The paper solves the problems of determining the asymptotics of the number of primes and the sums of functions of primes in a subset of the natural series that satisfies the conditions that the asymptotic density of the number of primes in this subset is constant and not equal to zero.
We prove two estimates for averages of sums of Kloosterman fractions over primes. The first of these improves previous results of Fouvry-Shparlinski and Baker.
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.
In this work we consider sums of primes that converging very slow. We set as a base, a reformulation of analytic prime number theorem and we use the values of Riemann Zeta function for the approximation. We also give the truncation error of these approximations
In this paper, we establish a theorem of Bombieri -- Vinogradov type for exponential sums over Piatetski-Shapiro primes $p= [n^{1/γ}]$ with $\frac{865}{886}<γ< 1$.
In this note we study Kloosterman sums twisted by a multiplicative characters modulo a prime power. We show, by an elementary calculation, that these sums become equidistributed on the real line with respect to a suitable measure.
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.
We obtain the general k-correlations for a short divisor sum related to primes.
Let $\mathbf H_2$ denote the set of even integers $n \not\equiv 1 \pmod 3$. We prove that when $H \ge X^{0.33}$, almost all integers $n \in \mathbf H_2$, $X < n \le X + H$ can be represented as the sum of a prime and the square of a prime. We also prove a similar result for sums of three squares of primes.
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 .
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…
A survey paper on some recent results on additive problems with prime powers.
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.
Under the assumption of the Riemann Hypothesis (RH), we prove explicit quantitative relations between hypothetical error terms in the asymptotic formulae for truncated mean-square average of exponential sums over primes and in the mean-square of primes in short intervals. We also remark that such relations are connected with a more precise form of Montgomery's pair-correlation conjecture.
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.
Improvements on exponential sums related to Piatetski-Shapiro primes
本文通过改进Type I和的方法，提高了与Piatetski-Shapiro素数相关的三次哥德巴赫问题的界，并研究了这些素数在算术级数和Beatty序列交集中的分布。
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。
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.
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.
We obtain the triple correlations for a truncated divisor sum related to primes. We also obtain the mixed correlations for this divisor sum when it is summed over the primes, and give some applications to primes in short intervals.
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.
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。
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.
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。
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.
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。
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。
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.
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…
In this paper we show that if $A$ is a subset of Chen primes with positive relative density $α$, then $A+A$ must have positive upper density at least $cαe^{-c^\prime\log(1/α)^{2/3}(\log\log(1/α))^{1/3}}$ in the natural numbers.
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…
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。
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。
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。
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。
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。
Assuming that the Generalized Riemann Hypothesis (GRH) holds, we prove an explicit formula for the number of representations of an integer as a sum of $k\geq 5$ primes. Our error terms in such a formula improve by some logarithmic factors an analogous result by Friedlander-Goldston.
For arbitrary $c_0>0$, if $A$ is a subset of the primes less than $x$ with cardinality $δx (\log x)^{-1}$ with $δ\geq (\log x)^{-c_0}$, then there exists a positive constant $c$ such that the cardinality of $A+A$ is larger than $c\, δx (\log\log x)^{-1}$.
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