Doychin Tolev
We establish Bombieri-Vinogradov's type result for the number of solutions of the ternary Goldbach problem with primes from arithmetic progressions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
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.
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.
Average orders of Goldbach Estimates in Arithmetic Progressions
本文研究了算术级数中Goldbach表示的平均数问题，获得了渐近结果，并证明了该结果本质上是最佳的。
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.
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.
We consider weighted averages of the number of representations of an even integer as a sum of two prime numbers, where each summand lies in a given arithmetic progression modulo a common integer $q$. Our result is uniform in a suitable range for $q$.
We investigate progressions in the set of pairs of integers $\mathbb{Z}^2$ and define a generalisation of the Jacobsthal function. For this function, we conjecture a specific upper bound and prove that this bound would be a sufficient condition for the truth of the Goldbach conjecture, the infinitude of prime twins, and more general of prime pairs with a fixed even difference.
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.
For two odd primes $p$ and $q$ such that $p<q$, let $A(p,q):=(a_k)_{k=1}^{\infty}$ be the arithmetic progression whose $k$th term is given by $a_k=(k-1)(q-p)+p$ (i.e., with $a_1=p$ and $a_2=q$). Here we conjecture that for every positive integer $a>1$ there exist a positive integer $n$ and two odd primes $p$ and $q$ such that $a$ can be expressed as a sum of the first $2n$ terms of the arithmetic progression $A(p,q)$. Notice that in the case of even $a$, this conjecture immediately follows from Goldbach's conjecture. We also propose the analogous conjecture for odd positive integers $a>1$ as w…
Assuming a conjecture on distinct zeros of Dirichlet L-functions we get asymptotic results on the average number of representations of an integer as the sum of two primes in arithmetic progression. On the other hand the existence of good error terms gives information on the the location of zeros of L-functions and possible Siegel zeros. Similar results are obtained for an integer in a congruence class expressed as the sum of two primes.
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.
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。
We formulate some refinements of Goldbach's conjectures based on heuristic arguments and numerical data. For instance, any even number greater than 4 is conjectured to be a sum of two primes with one prime being 3 mod 4. In general, for fixed $m$ and $a, b$ coprime to $m$, any positive even $n \equiv a + b \bmod m$ outside of a finite exceptional set is expected to be a sum of two primes $p$ and $q$ with $p \equiv a \bmod m$, $q \equiv b \bmod m$. We make conjectures about the growth of these exceptional sets.
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 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…
We prove that, for almost all $r \leq N^{1/2}/\log^{O(1)}N$, for any given $b_1 \mod r$ with $(b_1, r) = 1$, and for almost all $b_2 \mod r$ with $(b_2, r) = 1$, we have that almost all natural numbers $2n \leq N$ with $2n \equiv b_1 + b_2 \mod r$ can be written as the sum of two prime numbers $2n = p_1 + p_2$, where $p_1 \equiv b_1 \mod r$ and $p_2 \equiv b_2 \mod r$. This improves the previous result which required $r \leq N^{1/3}/\log^{O(1)}N$ instead of $r \leq N^{1/2}/\log^{O(1)}N$. We also improve some other results concerning variations of the problem.
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 …
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…
Primes in arithmetic progressions to large moduli, and Goldbach beyond the square-root barrier
本文利用三重可良好分解权重将素数分布水平提升至0.617，并在假设Selberg猜想下达到0.625，由此为孪生素数和Goldbach问题取得新上界，实现了Goldbach问题自1966年以来的最大改进。
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…
The exceptional set of Goldbach problem and Linnik's constant
该研究通过改进方法证明了Goldbach问题中例外集的上界，并给出了算术级数中最小素数的上界估计。
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…
A positive quadratic form is $(k,\ell)$-universal if it represents all the numbers $kx+\ell$ where $x$ is a non-negative integer, and almost $(k,\ell)$-universal if it represents all but finitely many of them. We prove that for any $k,\ell$ such that $k\nmid\ell$ there exists an almost $(k,\ell)$-universal diagonal ternary form. We also conjecture that there are only finitely many primes $p$ for which a $(p,\ell)$-universal diagonal ternary form exists (for any $\ell<p$) and we show the results of computer experiments that speak in favor of the conjecture.
In this note, assuming a variant of the Generalized Riemann Hypothesis, which does not exclude the existence of real zeros, we prove an asymptotic formula for the mean value of the representation function for the sum of two primes in arithmetic progressions. This is an improvement of the result of F. Rüppel in 2009, and the generalization of the result of A. Languasco and A. Zaccagnini concerning the ordinary Goldbach problem in 2012.
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…
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。
Jacobsthal's function was recently generalised for the case of paired progressions. It was proven that a specific bound of this function is sufficient for the truth of Goldbach's conjecture and of the prime pairs conjecture as well. We extended and adapted algorithms described for the computation of the common Jacobsthal function, and computed respective function values of the paired Jacobsthal function for primorial numbers for primes up to 73. All these values fulfil the conjectured specific bound. In addition to this note, we provide a detailed review of the algorithmic approaches and the c…
In this paper we improve the result of Akbary and Hambrook by a factor of log x by obtaining a better version of Vaughan's inequality and using the explicit variant of an inequality connected to the Möbius function, derived by Helfgott in his work on ternary Goldbach conjecture.
Arithmetic progressions represented by diagonal ternary quadratic forms
本文基于三元二次型理论，研究并证明了由L. Pehlivan、K. S. Williams和Z.-W. Sun提出的若干对角三元二次型具有表示特定等差数列的$(d,r)$-万有性。
We show that the exponent of distribution of the ternary divisor function $d_3$ in arithmetic progressions to prime moduli is at least 1/2+1/46, improving results of Heath-Brown and Friedlander--Iwaniec. Furthermore, when averaging over a fixed residue class, we prove that this exponent is increased to 1/2 +1/34.
The reverse Goldbach problem and a refined Zsiflaw--Legeis theorem
本文研究反素数的加法理论，证明了大奇数可以表示为素数与两个反素数之和，以及大奇数可以表示为两个素数与一个反素数之和，并证明几乎所有的偶数可以表示为一个素数与一个反素数之和，同时所有大整数可以表示为一个反素数与一个平方自由数之和。
In this paper we establish a function field analogue of a conjecture in number theory which is a combination of several famous conjectures, including the Hardy-Littlewood prime tuple conjecture, conjectures on the number of primes in arithmetic progressions and in short intervals, and the Goldbach conjecture. We prove an asymptotic formula for the number of simultaneous prime values of $n$ linear functions, in the limit of a large finite field.
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。
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。
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。
Schmidt's game is generally used to deduce qualitative information about the Hausdorff dimensions of fractal sets and their intersections. However, one can also ask about quantitative versions of the properties of winning sets. In this paper we show that such quantitative information has applications to various questions including: * What is the maximal length of an arithmetic progression on the "middle $ε$" Cantor set? * What is the smallest $n$ such that there is some element of the ternary Cantor set whose continued fraction partial quotients are all $\leq n$? * What is the Hausdorff dimens…
Large Value Estimates for Dirichlet Polynomials with Characters and Zero Density of Dirichlet $L$-Functions
研究狄利克雷多项式大值估计及狄利克雷\(L\)函数零点密度，通过改编Guth--Maynard方法改进指数，得到零点密度估计，进而推导模素数算术级数中最小哥德巴赫数新上界及短区间内算术级数中素数的新结果。
Exponential sums weighted by additive functions
本文研究了由加性函数加权的指数和，证明了其紧界，并应用Hardy-Littlewood圆法解决了与总素因子个数相关的Goldbach-Vinogradov三元问题的渐进行为。
Products of primes in arithmetic progressions
本文证明了Erdős素数乘积猜想的三元版本，通过建立素数特征和的乘性稠密模型定理，证明了任意充分大无立方因子整数模数的简化剩余类可表示为三个不超过该整数的素数之积。
Multiplicative functions in short arithmetic progressions
该研究探讨有界积性函数在短算术级数中的和的方差，改进推广了相关经典定理，给出近乎最优的例外模集合，并将结果应用于三素数乘积的Linnik型问题。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。