A. A. Glibichuk, S. V. Konyagin
Let $F_p$ be the field of a prime order $p$. Then for any positive integer $n>1$, for any $ε>0$, and for any subset $A$ of $F_p$, every element of $F_p$ can be represented as a sum of $N$ elements, each of them is a product of $n$ elements from $A$, where $N$ depends on $n$ and $\espilon$.
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Under the prime-tuple hypothesis, the set of signed primes is a sumset.
Best possible bounds are obtained for the concentration function of an additive arithmetic function on sequences of shifted primes.
A new explicit formula is proved for the contribution of the major arcs in the Goldbach and Generalized Twin Prime Problem, in which the level of the major arcs can be chosen very high. This will have many applications in the approximations of the mentioned problems.
The behaviour of a certain additive function in large intervals between consecutive primes
本文研究连续素数之间大区间内特定整数素因子所决定的加性函数的行为，结合素数大间隙相关结果、Erdős–Kac 定理及 Kubilius 模型得到主要结论。
Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, we establish a theorem of Bombieri-Vinogradov type for the Piatetski-Shapiro primes $p=[n^{1/γ}]$ with $\frac{85}{86}<γ<1$. Moreover, we use this result to prove that, for $0.9989445<γ<1$, there exist infinitely many Piatetski-Shapiro primes such that $p+2=\mathcal{P}_3$, which improves the previous results of Lu, Wang and Cai, and Peneva.
This paper discusses the additive prime divisor function $A(n) := \sum\limits_{p^α|| n} α\, p$ which was introduced by Alladi and Erd\H os in 1977. It is shown that $A(n)$ is uniformly distributed (mod $q$) for any fixed integer $q > 1.$
The main purpose of this survey is to introduce an inexperienced reader to additive prime number theory and some related branches of analytic number theory. We state the main problems in the field, sketch their history and the basic machinery used to study them, and try to give a representative sample of the directions of current research.
On the Weakly Prime-Additive Numbers with Length 4
本文在广义黎曼假设下证明了对任意正整数m，存在无穷多个能被m整除的长度为4的弱素数可加数，并给出了一个与长度为3情形类似的相关结果。
Let $E$ be a rational elliptic curve, and $K$ be an imaginary quadratic field. In this article we give a method to construct Heegner points when $E$ has a prime bigger than $3$ of additive reduction ramifying in the field $K$. The ideas apply to more general contexts, like constructing Darmon points attached to real quadratic fields which is presented in the appendix.
We continue investigations on the average number of representations of a large positive integer as a sum of given powers of prime numbers. The average is taken over a short interval, whose admissible length depends on whether or not we assume the Riemann Hypothesis.
Let $\mathtt{PRIMES}$ be the set of all primes. We show that a multiplicative function which satisfies \[ f(p+q-2) = f(p) + f(q) - f(2) \text{ for }p,q \in \mathtt{PRIMES} \] is one of the following: \begin{enumerate} \item $f$ is the identity function \item $f$ is the constant function with $f(n)=1$ \item $f(n)=0$ for $n \ge2$ unless $n$ is odd and squareful. \end{enumerate} As a consequence, a multiplicative function which satisfies \[ f(a+b) = f(a) + f(b) \text{ for }a,b \in \mathtt{PRIMES}-1 \] is the identity function.
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.
In [10] the third author of this paper presented two conjectures on the additive decomposability of the sequence of ''smooth'' (or ''friable'') numbers. Elsholtz and Harper [4] proved (by using sieve methods) the second (less demanding) conjecture. The goal of this paper is to extend and sharpen their result in three directions by using a different approach (based on the theory of $S$-unit equations).
Assume that $A\subseteq \Fp, B\subseteq \Fp^{*}$, $\1/4\leqslant\frac{|B|}{|A|},$ $|A|=p^α, |B|=p^β$. We will prove that for $p\geqslant p_0(β)$ one has $$\sum_{b\in B}E_{+}(A, bA)\leqslant 15 p^{-\frac{\min\{β, 1-α\}}{308}}|A|^3|B|.$$ Here $E_{+}(A, bA)$ is an additive energy between subset $A$ and it's multiplicative shift $bA$. This improves previously known estimates of this type.
On an additive problem involving fractional powers with one prime and an almost prime variables
本文研究分数次幂整数部分的二元加法表示，证明在指数范围内充分大整数可表为素数与素因子个数有界的殆素数之和，并改进已有结果。
Additive Problems with Primes from a Thin Bohr Set
本文研究满足薄Bohr集条件的素数上的加性问题，证明当τ∈(0,1/8)时该素数集合中存在无穷多个非平凡三项算术级数，并进一步考虑了二元Goldbach型问题。
In part I of this paper we studied additive decomposability of the set $\F_y$ of th $y$-smooth numbers and the multiplicative decomposability of the shifted set $\g_y=\F_y+\{1\}$. In this paper, focusing on the case of 'large' functions $y$, we continue the study of these problems. Further, we also investigate a problem related to the m-decomposability of $k$-term sumsets, for arbitrary $k$.
We show that if $E/\mathbb{Q}$ is an elliptic curve with a rational $p$-torsion for $p=2$ or $3$, then there is a congruence relation between Ramanujan's tau function and $E$ modulo $p$. We make use of such congruences to compute the Iwasawa invariants of $2$-adic and $3$-adic Mazur--Tate elements attached to Ramanujan's tau function. We also investigate numerically the Iwasawa invariants of the Mazur--Tate elements attached to an elliptic curve with additive reduction at a fixed prime number.
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。
Let $E$ be a rational elliptic curve and let $p$ be an odd prime of additive reduction. Let $K$ be an imaginary quadratic field and fix a positive integer $c$ prime to the conductor of $E$. The main goal of the present article is to define an anticyclotomic $p$-adic $L$-function $Ł$ attached to $E/K$ when $E/\QQ_p$ attains semistable reduction over an abelian extension. We prove that $Ł$ satisfies the expected interpolation properties; namely, we show that if $χ$ is an anticyclotomic character of conductor $cp^n$ then $χ(Ł)$ is equal (up to explicit constants) to $L(E,χ,1)$ or $L'(E,χ,1)$.
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。
We extend two results of Ruzsa and Vu on the additive complements of primes
Following an idea of Rowland we give a conjectural way to generate increasing sequences of primes using algorithms involving the gcd. These algorithms seem not so useless for searching primes since it appears we found sometime primes much more greater than the number of required iterations. In an other hand we propose new formulations of famous conjectures from the additive theory of numbers (the weak twin prime conjecture, the Polignac conjecture, the Goldbach conjecture or the very general Schinzel's hypothesis H). For the moment these are experimental results obtained using pari-gp.
On the Iwasawa Invariants of Mazur--Tate elements of elliptic curves at additive primes
该研究探讨有理数域上椭圆曲线在加法约化素数处Mazur–Tate元的λ-不变量，结合Sage计算数据给出加法潜在超奇异情形的实例与猜想，并将结果推广到相关尖点Hecke特征形式的对应不变量。
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。
Improving earlier estimates of several authors we show that the number E(X) of Goldbach exceptional even integers (that is, even integers which cannot be written as the sum of two primesw) below X satisfies tho bound E(X) < X^0.72 for sufficiently large X. In the proof crucial role is played by the explicit formula for the contribution of the major arcs proved in Part I of this series (arXiv:1804.05561). In this new version some misprints are corrected and an explanatory sentence is added to the common proof of theorems C and D.
This is an expository article to accompany my two lectures at the CDM conference. I have used this an excuse to make public two sets of notes I had lying around, and also to put together a short reader's guide to some recent joint work with T.Tao. Contents: 1. An exposition, without much detail, of the work of Goldston, Pintz and Yildirim on gaps between primes; 2. A detailed discussion of the work of Mauduit and Rivat establishing that 50 percent of the primes have odd digit sum when written in base 2; 3. A reader's guide to recent work of T.Tao and the author on linear equations in primes. T…
Fix $δ\in(0,1]$, $σ_0\in[0,1)$ and a real-valued function $\varepsilon(x)$ for which $\limsup_{x\to\infty}\varepsilon(x)\le 0$. For every set of primes ${\mathcal P}$ whose counting function $π_{\mathcal P}(x)$ satisfies an estimate of the form $$π_{\mathcal P}(x)=δ\,π(x)+O\bigl(x^{σ_0+\varepsilon(x)}\bigr),$$ we define a zeta function $ζ_{\mathcal P}(s)$ that is closely related to the Riemann zeta function $ζ(s)$. For $σ_0\le\frac12$, we show that the Riemann hypothesis is equivalent to the non-vanishing of $ζ_{\mathcal P}(s)$ in the region $\{σ>\frac12\}$. For every set of primes ${\mathcal …
This paper has been withdrawn by the author due to an error in the proof of Theorem 6.
We examine how closely a multiplicative function resembles an additive function. Given a multiplicative function $g$ and an additive function $f$, we examine the size of the quantity $E(f,g;x)=\# \{n\leq x:f(n)=g(n)\}$. We establish a lower bound for $E(Ω,g,x)$.
We investigate sumset decompositions of quite general sets with restricted prime factors. We manage to handle certain sets, such as the smooth numbers, even though they have little sieve amenability, and conclude that these sets cannot be written as a ternary sumset. This proves a conjecture by Sárközy. We also clean up and sharpen existing results on sumset decompositions of the prime numbers.
The study of the additive volume of sets can be reduced to the case of one-dimensional sets. The exact values of the volume of extremal sets are given as a conjecture.
In this paper we extend and improve our results on weighted averages for the number of representations of an integer as a sum of two powers of primes. Let $1\le \ell_1 \le \ell_2$ be two integers, $Λ$ be the von Mangoldt function and % \(r_{\ell_1,\ell_2}(n) = \sum_{m_1^{\ell_1} + m_2^{\ell_2}= n} Λ(m_1) Λ(m_2) \) % be the weighted counting function for the number of representation of an integer as a sum of two prime powers. Let $N \geq 2$ be an integer. We prove that the Cesàro average of weight $k > 1$ of $r_{\ell_1,\ell_2}$ over the interval $[1, N]$ has a development as a sum of terms depe…
Following the sum-product paradigm, we prove that for a set $B$ with polynomial growth, the product set $B.B$ cannot contain large subsets with size of order $|B|^2$ with small doubling. It follows that the additive energy of $B.B$ is asymptotically $o(|B|^6)$. In particular, we extend to sets of small doubling and polynomial growth the classical Multiplication Table theorem of Erdős saying that $|[1..n]. [1..n]| = o(n^2)$.
We show that a set is almost periodic if and only if the associated exponential sum is concentrated in the minor arcs. Hence binary additive problems involving almost periodic sets can be solved using the circle method.
A lower bound for additive complements to the primes
研究满足特定条件的集合大小，得到素数加法补集的一个下界，给出Ruzsa猜想的量化形式，排除了Erdős问题32中提及的特定大小加法补集，还构造了相关连续整数块。
A survey paper on some recent results on additive problems with prime powers.
Joint distribution in residue classes of families of polynomially-defined additive functions
本文研究了多项式定义的加性函数族在剩余类中的联合分布，在近乎最优的假设下证明了其联合等分布，从而得到了变化模下 Siegel-Walfisz 定理的类似结果。
We prove that finite sets of real numbers satisfying $|AA| \leq |A|^{1+ε}$ with sufficiently small $ε> 0$ cannot have small additive bases nor can they be written as a set of sums $B+C$ with $|B|, |C| \geq 2$. The result can be seen as a real analog of the conjecture of Sárközy that multiplicative subgroups of finite fields of prime order are additively irreducible.
We introduce the notion of saturated sets of primes of an algebraic number field and prove an analogue of Riemann's existence theorem for the decomposition groups of infinite stably saturated sets of primes.
An MSTD set is a finite set of integers with more sums than differences. It is proved that, for infinitely many positive integers $k$, there are infinitely many affinely inequivalent MSTD sets of cardinality $k$. There are several related open problems.
We study the solubility of the binary additive equation $[m^c] + [p^c] = n$, where $m$ is an integer, $p$ is a prime number, and $c$ is a fixed real number in the range $1 < c < 3/2$.
In this paper, we study the hybrid problem of Hua's theorem and the Piatetski-Shapiro prime number theorem, and obtain results in this direction of the nonhomogeneous case $k=3$, which deepen the classical result of Hua.
On a problem of minimal additive complements for not eventually periodic $S$-difference sets
针对非最终周期S-差集的极小加法补集问题，本文对Ma和Chen提出的一个相关问题给出了肯定回答。
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。
On an average ternary problem with prime powers
本文研究了三次加法问题中素数幂的平均值，并延续了近期相关研究。
Dynamics on the number of prime divisors for additive arithmetic semigroups
本文在有限域算术半群中推广了Bergelson和Richter的定理，并探讨了整数最大素因子分布的动态性质。
Dynamical generalizations of the Prime Number Theorem and disjointness of additive and multiplicative semigroup actions
本研究建立两个蕴含多项乘性数论经典结果的遍历定理，将Sarnak的Möbius不交性猜想重构于新动力学框架，提出聚焦加乘半群作用不交性的猜想扩展并证明其若干特例。
We investigate the problem of r almost-primes represented by sets of quadratic forms and give upper bounds for r. Our results extend work of Diamond and Halberstam in which they investigated the corresponding problem for polynomials.
Additive and multiplicative densities, prime valuations and symbolic models
研究自然数集子集沿规定福勒序列的加法与乘法密度，证明加法上密度为1时沿合适乘法福勒序列乘法密度为1及独立性，通过素数赋值坐标得到密度公式和随机模型，还将乘法相关性应用于符号动力系统并给出遍历性和混合准则。
Triangular and tetrahedral number differences of sumset sizes in additive number theory
本文研究有限整数集合sumset大小的完整范围，发现在大小为4的集合流行sumset大小分布中存在与三角数和四面体数相关的意外模式。
Let $x$ be a positive integer. We give an asymptotic formula for the number of primes in the set $\{\fl{x/n}, 1 \le n \le x\}$ and give some related results.
An inverse theorem on sets with rich additive structure modulo primes
本文研究了模素数下具有丰富加法结构集合的反向定理，证明了相关结构结果并改进了较大筛法界。
Let $p\equiv1\pmod3$ be a prime . We study several topics on additive decompositions concerning the set $C_p$ of all non-zero cubes in the finite field of $p$ elements. For example, we show that when $p>184291$ , the set $C_p$ has no decomposition of the form $C_p=A+B+C$ with $|A|,|B|,|C|\ge2$.
Sparse sets that satisfy the prime number theorem
本文研究满足素数定理的稀疏集合，其中形如 $\lfloor x/n^t \rfloor$ 的集合在 $t$ 趋于无穷时密度趋近 $1/x$。
Let [t] be the integral part of the real number t and let 1 P be the characteristic function of the primes. Denote by $π$ G (x) the number of primes in the floor function set G(x) := {[ x n ] : 1 n x} and by S 1 P (x) the number of primes in the sequence {[ x n ]} n 1. Very recently, Heyman proves
When investigating the distribution of the Euler totient function, one encounters sets of primes P where if p is in P then r is in P for all r|(p-1). While it is easy to construct finite sets of such primes, the only infinite set known is the set of all primes. We translate this problem into the function field setting and construct an infinite such set in F_p[x] whenever p is equivalent to 2 or 5 modulo 9.
In the present paper we investigate distributional properties of sparse sequences modulo almost all prime numbers. We obtain new results for a wide class of sparse sequences which in particular find applications on additive problems and the discrete Littlewood problem related to lower bound estimates of the $L_1$-norm of trigonometric sums.
We show that if A is a subset of {1, ..., n} such that it has no pairs of elements whose difference is equal to p-1 with p a prime number, then the size of A is O(n(loglog n)^(-clogloglogloglog n)) for some positive constant c.
We study the concentration of the distribution of an additive function, when the sequence of prime values of $f$ decays fast and has good spacing properties. In particular, we prove a conjecture by Erdos and Katai on the concentration of $f(n)=\sum_{p|n}(\log p)^{-c}$ when $c>1$.
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