Jing Run Chen proved in 1966 that $p+2$ has at most two prime factors for infinitely many primes $p$. However, due to the parity problem we do not know whether $p+2$ has an odd (or even) number of prime factors infinitely often. In the present work it is proved that $p+d$ has an odd number of prime factors for at least one value of d=2,4,...16.
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.
数值证据表明，只有约$2\%$对$p,p+2$孪生素数，$p+2$有更多的原根比$p$。如果发生这种情况，我们就说$p$是例外的（只有两个例外对与$5 \leq p \leq 10{,}000$）。在Bateman-Horn猜想的假设下，证明了孪生素数对的至少$0.47\%$是例外的，至少$65.13\%$不是例外的.我们还推测了特殊孪生素数比例的精确公式。
We improve Bombieri's asymptotic sieve to localise the variables. As a consequence, we prove, under a Elliott-Halberstam conjecture, that there exists an infinity of twins almost prime. Those are prime numbers $p$ such that for all $\varepsilon>0$, $p-2$ is either a prime number or can be written as $p_1 p_2$ where $p_1$ and $p_2$ are prime and $p_1<X^{\varepsilon}$, and we give the explicit asymptotic.
We present the geometry lying behind counting twin prime polynomials in $\mathbb{F}_q[T]$ in general. We compute cohomology and explicitly count points by means of a twisted Lefschetz trace formula applied to these parametrizing varieties for cubic twin prime polynomials. The elliptic curve $X^3 = Y(Y-1)$ occurs in the geometry, and thus counting cubic twin prime polynomials involves the associated modular form. In theory, this approach can be extended to higher degree twin primes, but the computations become harder. The formula we get in degree $3$ is compatible with the Hardy-Littlewood heur…
筛被构造用于第一类孪生素数，其形式为2 m +/-D，D>=3奇数。它们的特征在于它们的双D-I秩m。他们没有平价问题。使用奇素数p>=5来识别和计数非秩数。Twin-D-I秩和非秩构成正整数的集合。非秩的正则性允许获得双D-I秩的数量。它涉及相当多的取消，使渐近形式的主要条款崩溃的预期形式，但其系数取决于D。
We present an algorithm analogous to the sieve of Eratosthenes that produces the list of twin primes. Next, we count the number of twin primes resulting from the construction with two different heuristic arguments. The first method is essentially the same as the one in Hardy and Wright. However, the second method is novel. It results in the same asymptotic formula but it uses a simpler correction factor than theirs. Though we have no theory for the accuracy of our estimates, we compute them both without and with the correction factor and they turn out to be close to the actual counts up to 800…
Garcia, Kahoro, and Luca showed that the Bateman-Horn conjecture implies $ϕ(p-1) \geq ϕ(p+1)$ for a majority of twin-primes pairs $p,p+2$ and that the reverse inequality holds for a small positive proportion of the twin primes. That is, $p$ tends to have more primitive roots than does $p+2$. We prove that Dickson's conjecture, which is much weaker than Bateman-Horn, implies that the quotients $\frac{ϕ(p+1)}{ϕ(p-1)}$, as $p,p+2$ range over the twin primes, are dense in the positive reals. We also establish several Schinzel-type theorems, some of them unconditional, about the behavior of $\frac{…
We consider the exceptional set in the binary Goldbach problem for sums of two almost twin primes. Our main result is a power-saving bound for the exceptional set in the problem of representing $m=p_1+p_2$ where $p_1+2$ has at most $2$ prime divisors and $p_2+2$ has at most $3$ prime divisors. There are three main ingredients in the proof: a new transference principle like approach for sieves, a combination of the level of distribution estimates of Bombieri--Friedlander--Iwaniec and Maynard with ideas of Drappeau to produce power savings, and a generalisation of the circle method arguments of …
We say that a prime number $p$ is an $\textit{Artin prime}$ for $g$ if $g$ mod $p$ generates the group $(\mathbb{Z}/p\mathbb{Z})^{\times}$. For appropriately chosen integers $d$ and $g$, we present a conjecture for the asymptotic number $π_{d,g}(x)$ of primes $p \leq x$ such that both $p$ and $p+d$ are Artin primes for $g$. In particular, we identify a class of pairs $(d,g)$ for which $π_{d,g}(x) =0$. Our results suggest that the distribution of Artin prime pairs, amongst the ordinary prime pairs, is largely governed by a Poisson binomial distribution.
We study the distribution of prime numbers under the unlikely assumption that Siegel zeros exist. In particular we prove for \[ \sum_{n \leq X} Λ(n) Λ(\pm n+h) \] an asymptotic formula which holds uniformly for $h = O(X)$. Such an asymptotic formula has been previously obtained only for fixed $h$ in which case our result quantitatively improves those of Heath-Brown (1983) and Tao and Teräväinen (2021). Since our main theorems work also for large $h$ we can derive new results concerning connections between Siegel zeros and the Goldbach conjecture and between Siegel zeros and primes in almost al…
The transference principle of Green and Tao enabled various authors to transfer Szemerédi's theorem on long arithmetic progressions in dense sets to various sparse sets of integers, mostly sparse sets of primes. In this paper, we provide a transference principle which applies to general affine-linear configurations of finite complexity. We illustrate the broad applicability of our transference principle with the case of almost twin primes, by which we mean either Chen primes or "bounded gap primes", as well as with the case of primes of the form $x^2+y^2+1$. Thus, we show that in these sets of…
Using geometric methods, we improve on the function field version of the Burgess bound, and show that, when restricted to certain special subspaces, the Möbius function over $\mathbb F_q[T]$ can be mimicked by Dirichlet characters. Combining these, we obtain a level of distribution close to $1$ for the Möbius function in arithmetic progressions, and resolve Chowla's $k$-point correlation conjecture with large uniformity in the shifts. Using a function field variant of a result by Fouvry-Michel on exponential sums involving the Möbius function, we obtain a level of distribution beyond $1/2$ for…
Computer experiments reveal that twin primes tend to center on nonsquarefree multiples of 6 more often than on squarefree multiples of 6 compared to what should be expected from the ratio of the number of nonsquarefree multiples of 6 to the number of squarefree multiples of 6 equal $π^2/3-1$, or ca 2.290. For multiples of 6 surrounded by twin primes, this ratio is 2.427, a relative difference of ca $6.0\%$ measured against the expected value. A deviation from the expected value of this ratio, ca $1.9\%$, exists also for isolated primes. This shows that the distribution of primes is biased towa…
构造了一个距离为4的孪生素数的筛子，它们的形式为3（2 m +1）+/-2，其特征在于它们的孪生-4秩为2 m +1。它没有平价问题。非秩被标识为所有其他奇数，并使用奇素数p>=5计数。双4秩和非秩构成奇数的集合。非等级的规则允许收集关于它们的信息以获得双4等级的数量的Legendre类型的总和。由于其中有相当多的消去项，其主项的渐近律具有其系数的期望形式和大小。
