Hongze Li, Hao Pan
We prove that there exists a k_0>0 such that every sufficiently large odd integer n with 3\mid n can be represented as p_1+p_2+p_3, where p_1,p_2 are Chen's primes and p_3 is a prime with p_3+2 has at most k_0 prime factors.
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We establish Bombieri-Vinogradov's type result for the number of solutions of the ternary Goldbach problem with primes from arithmetic progressions.
We describe a computation that confirms the ternary Goldbach Conjecture up to 8,875,694,145,621,773,516,800,000,000,000 (>8.875e30).
We consider the ternary Goldbach problem with two prime variables of the form $k^2+m^2+1$ and find an asymptotic formula for the number of its solutions.
Let $η$ be a quadratic irrationality. The variant of a ternary problem of Goldbach involving primes such that $a<\{ηp\}<b$, where $a$ and $b$ are arbitrary numbers of the interval $(0,1)$, solved in this paper.
In this paper, we proved a theorem that every large enough odd number can be represented as the sum of three almost equal Piatetski-Shapiro primes.
We give a more comrehensive treatment of Chen's double sieve and improve related constants in Goldbach's conjecture and the twin prime problem.
Almost all primes are not needed in Ternary Goldbach
该研究针对三元哥德巴赫猜想，通过可容许同余扩张系统构造素数集$\u2119$，证明即便剔除几乎所有素数，仅用$\u2119$中的素数仍可满足该猜想，深化了对素数加法性质的认识。
In the present paper we prove that under the assumption of the GRH (Generalized Riemann Hypothesis) each sufficiently large odd integer can be expressed as the sum of a prime and two isolated primes.
The ternary Goldbach conjecture (or three-prime conjecture) states that every odd number greater than 5 can be written as the sum of three primes. The purpose of this book is to give the first proof of the conjecture, in full.
The ternary Goldbach conjecture, or three-primes problem, asserts that every odd integer $n$ greater than $5$ is the sum of three primes. The present paper proves this conjecture. Both the ternary Goldbach conjecture and the binary, or strong, Goldbach conjecture had their origin in an exchange of letters between Euler and Goldbach in 1742. We will follow an approach based on the circle method, the large sieve and exponential sums. Some ideas coming from Hardy, Littlewood and Vinogradov are reinterpreted from a modern perspective. While all work here has to be explicit, the focus is on qualita…
In the present paper we prove that every sufficiently large odd integer $N$ can be represented in the form \begin{equation*} N=p_1+p_2+p_3\,, \end{equation*} where $p_1,p_2,p_3$ are primes, such that $p_1=x^2 + y^2 +1$, $p_2=[n^c]$.
We prove an inverse ternary Goldbach-type result. Let $N$ be sufficiently large and $c>0$ be sufficiently small. If $A_1,A_2,A_3\subset [N]$ are subsets with $|A_1|,|A_2|,|A_3|\geq N^{1/3-c}$, then $A_1+A_2+A_3$ contains a composite number. This improves on the bound $N^{1/3}$ from Gallagher's larger sieve. The main ingredients in our argument include a type of inverse sieve result in the larger sieve regime, and a variant of the analytic large sieve inequality.
For every even integer N, denote by D_{1,2}(N) the number of representations of N as a sum of a prime and an integer having at most two prime factors. In this paper, we give a new lower bound for D_{1,2}(N).
For A,epsilon>0 and any sufficiently large odd n we show that for almost all k up to n^{1/5-epsilon} there exists a representation n=p1+p2+p3 with primes in residue classes b1,b2,b3 mod k for almost all admissible triplets b1,b2,b3 of reduced residues mod k.
Let $\mathcal{P}$ denote the set of all primes. $P_{1},P_{2},P_{3}$ are three subsets of $\mathcal{P}$. Let $\underlineδ(P_{i})$ $(i=1,2,3)$ denote the lower density of $P_{i}$ in $\mathcal{P}$, respectively. It is proved that if $\underlineδ(P_{1})>5/8$, $\underlineδ(P_{2})\geq5/8$, and $\underlineδ(P_{3})\geq5/8$, then for every sufficiently large odd integer n, there exist $p_{i} \in P_{i}$ such that $n=p_{1}+p_{2}+p_{3}$. The condition is the best possible.
Density ternary Goldbach for primes in a fixed residue class
本文证明，在同余 1 mod 3 的素数中，相对密度超过 1/2 的子集可表示所有充分大且同余 0 mod 3 的奇数为三元素数和，并证明该密度阈值最优。
Let $$γ^*=\frac{8}{9}+\frac{2}{3}\:\frac{\log(10/9)}{\log 10}\:(\approx 0.919\ldots)\:.$$ Let $γ^*<γ_0\leq 1$, $c_0=1/γ_0$ be fixed. Let also $a_0\in\{0,1,\ldots, 9\}$.\\ We prove on assumption of the Generalized Riemann Hypothesis that each sufficiently large odd integer $N_0$ can be represented in the form $$N_0=p_1+p_2+p_3\:,$$ where the $p_i$ are of the form $p_i=[n_i^{c_0}]$, $n_i\in\mathbb{N}$, for $i=1,2$ and the decimal expansion of $p_3$ does not contain the digit $a_0$.\\ The proof merges methods of J. Maynard from his paper on the infinitude of primes with restricted digits, results…
We show that for every fixed $A>0$ and $θ>0$ there is a $\vartheta=\vartheta(A,θ)>0$ with the following property. Let $n$ be odd and sufficiently large, and let $Q_{1}=Q_{2}:=n^{\h}(\log n)^{-\vartheta}$ and $Q_{3}:=(\log n)^θ$. Then for all $q_{3}\leq Q_{3}$, all reduced residues $a_{3}$ mod $q_{3}$, almost all $q_{2}\leq Q_{2}$, all admissible residues $a_{2}$ mod $q_{2}$, almost all $q_{1}\leq Q_{1}$ and all admissible residues $a_{1}$ mod $q_{1}$, there exists a representation $n=p_{1}+p_{2}+p_{3}$ with primes $p_{i}\equiv a_{i} (q_{i})$, $i=1,2,3$.
The Exceptional Set in Goldbach's Problem with two Chen Primes
该研究针对满足$n\equiv4\pmod{6}$的自然数，通过结合Bombieri–Vinogradov定理的幂节省变体的高效筛策略，改进了哥德巴赫问题中两个陈素数和的相关结论，确定了其例外集的幂节省性质。
The ternary Goldbach conjecture, or three-primes problem, states that every odd number $n$ greater than $5$ can be written as the sum of three primes. The conjecture, posed in 1742, remained unsolved until now, in spite of great progress in the twentieth century. In 2013 -- following a line of research pioneered and developed by Hardy, Littlewood and Vinogradov, among others -- the author proved the conjecture. In this, as in many other additive problems, what is at issue is really the proper usage of the limited information we possess on the distribution of prime numbers. The problem serves a…
On Chen's theorem, Goldbach's conjecture and almost prime twins II
本文证明了关于Goldbach猜想和孪生素数问题的筛法结果，将素数表示的下界渐近常数提升至1.9728，接近Hardy-Littlewood猜想的渐近常数2，结合了加权筛法、Chen转换原理等多种筛法技术。
Let $$γ^*:=\frac{8}{9}+\frac{2}{3}\:\frac{\log(10/9)}{\log 10}\:(\approx 0.919\ldots)\:,\ γ^*<\frac{1}{c_0}\leq 1\:.$$ Let $γ^*<γ_0\leq 1$, $c_0=1/γ_0$ be fixed. Let also $a_0\in\{0,1,\ldots, 9\}$. In [23] we proved on assumption of the Generalized Riemann Hypothesis (GRH), that each sufficiently large odd integer $N_0$ can be represented in the form $$N_0=p_1+p_2+p_3\:,$$ where for $i=2, 3$ the primes $p_i$ are Piatetski-Shapiro primes - primes of the form $p_i=[n_i^{c_0}]$, $n_i\in\mathbb{N}$ - whereas the decimal expansion of $p_1$ does not contain the digit $a_0$. In this paper we replace …
Some mean value theorems in the style of Bombieri-Vinogradov's theorem are discussed. They concern binary and ternary additive problems with primes in arithmetic progressions and short intervals. Nontrivial estimates for some of these mean values are given. As application inter alia, we show that for large odd n\not\equiv 1 (6), Goldbach's ternary problem n=p_1+p_2+p_3 is solvable with primes p_1,p_2 in short intervals p_i \in [X_i,X_i+Y] with X_{i}^{θ_i}=Y, i=1,2, and θ_1,θ_2\geq 0.933 such that (p_1+2)(p_2+2) has at most 9 prime factors.
The ternary Goldbach conjecture states that every odd number n>=7 is the sum of three primes. The estimation of sums of the form \sum_{p\leq x} e(αp), α= a/q + O(1/q^2), has been a central part of the main approach to the conjecture since (Vinogradov, 1937). Previous work required q or x to be too large to make a proof of the conjecture for all n feasible. The present paper gives new bounds on minor arcs and the tails of major arcs. This is part of the author's proof of the ternary Goldbach conjecture. The new bounds are due to several qualitative improvements. In particular, this paper presen…
The ternary Goldbach conjecture states that every odd number $n\geq 7$ is the sum of three primes. The estimation of the Fourier series $\sum_{p\leq x} e(αp)$ and related sums has been central to the study of the problem since Hardy and Littlewood (1923). Here we show how to estimate such Fourier series for $α$ in the so-called major arcs, i.e., for $α$ close to a rational of small denominator. This is part of the author's proof of the ternary Goldbach conjecture. In contrast to most previous work on the subject, we will rely on a finite verification of the Generalized Riemann Hypothesis up to…
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…
We study the Goldbach problem for primes represented by the polynomial $x^2+y^2+1$. The set of such primes is sparse in the set of all primes, but the infinitude of such primes was established by Linnik. We prove that almost all even integers $n$ satisfying certain necessary local conditions are representable as the sum of two primes of the form $x^2+y^2+1$. This improves a result of Matomäki, which tells that almost all even $n$ satisfying a local condition are the sum of one prime of the form $x^2+y^2+1$ and one generic prime. We also solve the analogous ternary Goldbach problem, stating tha…
The Goldbach conjecture states that every even number can be decomposed as the sum of two primes. Let $D(N)$ denote the number of such prime decompositions for an even $N$. It is known that $D(N)$ can be bounded above by $$ D(N) \leq C^* Θ(N), \quad Θ(N):= \frac{N}{\log^2 N}\prod_{\substack{p|N p>2}} \left( 1 + \frac{1}{p-2}\right)\prod_{p>2}\left(1-\frac{1}{(p-1)^2}\right) $$ where $C^*$ denotes Chen's constant. It is conjectured that $C^*=2$. In 2004, Wu showed that $C^* \leq 7.8209$. We attempted to replicate his work in computing Chen's constant, and in doing so we provide an improved appr…
With the uniform positions we prove theorems of Landau and Hardy-Littlwood type for Goldbach, Chen, Lemoine-Levy and other binary partitions of positive integers. We also pose some new conjectures.
Multiple sums with the Möbius function
本文研究含 Möbius 函数的多重双线性和，对广泛方程的解建立非平凡界，得到三元 Goldbach 型结果，并在二元情形通过限制第三变量取得部分进展。
We prove some general recursions for the numbers of representations of positive integers as a sum x+y, x in X, y in Y, where X,Y are increasing sequences. In particular, we obtain recursions for the number of the Goldbach, Lemoine-Levy, Chen and other binary partitions.
We prove that if $x$ is large enough, namely $x\ge x_0$, then there exists a prime between $x(1- Δ^{-1})$ and $x$, where $Δ$ is an effective constant computed in terms of $x_0$. This improves some previous results of Ramaré and Saouter and is used to verify the Ternary Goldbach Conjecture for the integers up to $7.86\cdot 10^{27}$.
Theorem $(1+1.9)$ on the Goldbach Conjecture
本文通过构造新的加权筛法和采用新的分析工具，无条件证明了命题 $(1+1.9)$ 成立，即在哥德巴赫猜想中，每个充分大的偶数可表示为 $p+rq$ 且 $r \leq q^{0.9}$，其中 $r$ 为1或素数，$p,q$ 为素数；在埃利奥特-哈尔伯斯坦猜想下指数可改进至1.4，同时类似结果应用于孪生素数猜想。
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。
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