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 8009.
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 suggest other models of sieve generated sequences like the Sieve of Eratosthenes to explain randomness properties of the prime numbers, like the twin prime conjecture, the lim sup conjecture, the Riemann conjecture, and the prime number theorem.
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。
In this paper we consider a slightly different sieve method from Eratosthenes' to get primes. We find the periodicity and mirror symmetry of the pattern.
We make two algorithms that generate all prime numbers up to a given limit, they are a development of sieve of Eratosthenes algorithm, we use two formulas to achieve this development, where all the multiples of prime number 2 are eliminated in the first formula, and all the multiples of prime numbers 2 and 3 are eliminated in the second formula. Using the first algorithm we proof sieve of Sundaram's algorithm, then we improve it to be more efficient prime generating algorithm. We will show the difference in performance between all the algorithms we will make and sieve of Eratosthenes algorithm…
A few years ago we identified a recursion that works directly with the gaps among the generators in each stage of Eratosthenes sieve. This recursion provides explicit enumerations of sequences of gaps among the generators, which sequences are known as constellations. By studying this recursion on the cycles of gaps across stages of Eratosthenes sieve, we are able to provide evidence on a number of open problems regarding gaps between prime numbers. The basic counts of short constellations in the cycles of gaps provide evidence toward the twin prime conjecture and toward resolving a series of q…
This paper is motivated by the following question in sieve theory. Given a subset $X\subset [N]$ and $α\in (0,1/2)$. Suppose that $|X\pmod p|\leq (α+o(1))p$ for every prime $p$. How large can $X$ be? On the one hand, we have the bound $|X|\ll_αN^α$ from Gallagher's larger sieve. On the other hand, we prove, assuming the truth of an inverse sieve conjecture, that the bound above can be improved (for example, to $|X|\ll_αN^{O(α^{2014})}$ for small $α$). The result follows from studying the average size of $|X\pmod p|$ as $p$ varies, when $X=f(\mathbb{Z})\cap [N]$ is the value set of a polynomial…
We improve Irving's method of the double-sieve by using the DHR sieve. By extending the upper and lower sieve functions into their respective non-elementary ranges, we are able to make improvements on the previous records on the number of prime factors of irreducible polynomials at prime arguments. In particular, we prove that irreducible quadratics over $\mathbb{Z}$ satisfying necessary local conditions are $P_4$ infinitely often.
We study certain aspects of the Selberg sieve, in particular when sifting by rather thin sets of primes. We derive new results for the lower bound sieve suited especially for this setup and we apply them in particular to give a new sieve-propelled proof of Linnik's theorem on the least prime in an arithmetic progression in the case of the presence of exceptional zeros.
Let $k\geq 2$ and $\mathcal{P} (n) = (A_1 n + B_1 ) \cdots (A_k n + B_k)$ where all the $A_i, B_i$ are integers. Suppose that $\mathcal{P} (n)$ has no fixed prime divisors. For each choice of $k$ it is known that there exists an integer $\varrho_k$ such that $\mathcal{P} (n)$ has at most $\varrho_k$ prime factors infinitely often. We used a new weighted sieve set-up combined with a device called an $\varepsilon$-trick to improve the possible values of $\varrho_k$ for $k\geq 7$. As a by-product of our approach, we improve the conditional possible values of $\varrho_k$ for $k\geq 4$, assuming th…
A prime sieve is an algorithm that finds the primes up to a bound $n$. We say that a prime sieve is incremental, if it can quickly determine if $n+1$ is prime after having found all primes up to $n$. We say a sieve is compact if it uses roughly $\sqrt{n}$ space or less. In this paper we present two new results: (1) We describe the rolling sieve, a practical, incremental prime sieve that takes $O(n\log\log n)$ time and $O(\sqrt{n}\log n)$ bits of space, and (2) We show how to modify the sieve of Atkin and Bernstein (2004) to obtain a sieve that is simultaneously sublinear, compact, and incremen…
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 …
Almost primes and primes that are sums of two squares plus 1
Institut Élie Cartan de Lorraine, Université de Lorraine, CNRS（洛林埃利嘉当研究所，洛林大学，法国国家科学研究中心）；Department of Mathematics, University of Illinois Urbana–Champaign（伊利诺伊大学厄巴纳-香槟分校数学系）；Max-Planck-Institut für Mathematik（马克斯·普朗克数学研究所）
We complement the argument of M. Z. Garaev (2009) with several other ideas to obtain a stronger version of the large sieve inequality with sparse exponential sequences of the form $λ^{s_n}$. In particular, we obtain a result which is non-trivial for monotonically increasing sequences $\cal{S}=\{s_n \}_{n=1}^{\infty}$ provided $s_n\le n^{2+o(1)}$, whereas the original argument of M. Z. Garaev requires $s_n \le n^{15/14 +o(1)}$ in the same setting. We also give an application of our result to arithmetic properties of integers with almost all digits prescribed.
The method of proof of Balog and Ruzsa and the large sieve of Linnik are used to investigate the behaviour of the $L^{1}$ norm of a wide class of exponential sums over the square-free integers and the primes. Further, a new proof of the lower bound due to Vaughan for the $L^{1}$ norm of an exponential sum with the von Mangoldt $Λ$ function over the primes is furnished. Ramanujan's sum arises naturally in the proof, which also employs Linnik's large sieve.
Most prime gaps results have been proven using tools from analytic or algebraic number theory in the last few centuries. In this paper, we would like to present some probabilistic way of proving many essential results. A major component of the proof is a probabilistic approach to the sieve method. In addition, we discuss their connections with recent work by Zhang and Maynard on small and large gaps in prime numbers.
A new Combined Sieve algorithm is presented with cost proportional to the number of enumerated factors over a series of intervals. This algorithm achieves a significant speedup, over a traditional sieve, when handling many ([10^4, 10^7]) intervals concurrently. The speedup comes from a space-time tradeoff and a novel solution to a modular equation. In real world tests, this new algorithm regularly runs 10,000x faster. This faster sieve paired with higher sieving limits eliminates more composites and accelerates the search for large prime gaps by 30-70%. During the development and testing of th…
A few years ago we identified a recursion that works directly with the gaps among the generators in each stage of Eratosthenes sieve. This recursion provides explicit enumerations of sequences of gaps among the generators, which sequences are known as constellations. The populations of gaps and constellations across stages of Eratosthenes sieve are modeled exactly by discrete dynamic systems. These models and their asymptotic behaviors provide evidence on a number of open problems regarding gaps between prime numbers. For Eratosthenes sieve we show that the analogue of Polignac's conjecture is…
We derive explicit formulas for integrals of certain symmetric polynomials used in Keiju Sono's multidimensional sieve of $E_2$-numbers, i.e., integers which are products of two distinct primes. We use these computations to produce the currently best-known bounds for gaps between multiple $E_2$-numbers. For example, we show there are infinitely many occurrences of four $E_2$-numbers within a gap size of 94 unconditionally and within a gap size of 32 assuming the Elliott-Halberstam conjecture for primes and sifted $E_2$-numbers.
We study an instance of the Affine Sieve, producing a level of distribution beyond that which can be obtained from current techniques, even assuming a Selberg/Ramanujan-type spectral gap. In particular, we consider the set of hypotenuses in a thin orbit of Pythagorean triples. Previous work [Kon07, Kon09, KO12] gave an exponent of distribution alpha < 1/12 coming from Gamburd's [Gam02] gap theta = 5/6, thereby producing R = 13 almost primes in this linear sieve problem (see Sec. 1 for definitions). If conditioned on a best possible gap theta = 1/2, the known method would give an exponent alpha…
A few years ago we identified a recursion that works directly with the gaps among the generators in each stage of Eratosthenes sieve. This recursion provides explicit enumerations of sequences of gaps among the generators, which sequences are known as constellations. Over the last year we identified a discrete linear system that exactly models the population of any gap across all stages of the sieve. In August 2014 we summarized our results from analyzing this discrete model on populations of single gaps. This paper extends the discrete system to model the populations of constellations of gaps…
It is proven that there are infinitely prime pairs whose difference is no greater than 84. In proving that result, a new sieve weight is constructed. The new idea is to construct the largest sieve support which admits the use of the Bombieri-Vinogradov theorem.
We prove large sieve inequalities with multivariate polynomial moduli and deduce a general Bombieri--Vinogradov type theorem for a class of polynomial moduli having a sufficient number of variables compared to its degree. This sharpens previous results of the first author in two aspects: the range of the moduli as well as the class of polynomials which can be handled. As a consequence, we deduce that there exist infinitely many primes $p$such that $p-1$ has a prime divisor of size $\gg p^{2/5+o(1)}$ that is the value of an incomplete norm form polynomial.
We establish new mean value theorems for primes of size $x$ in arithmetic progressions to moduli as large as $x^{3/5-ε}$ when summed with suitably well-factorable weights. This extends well-known work of Bombieri, Friedlander and Iwaniec, who handled moduli of size at most $x^{4/7-ε}$. This has consequences for the level of distribution for sieve weights coming from the linear sieve.
Recently Oliver and Soundararajan made conjectures based on computational enumerations about the frequency of occurrence of pairs of last digits for consecutive primes. By studying Eratosthenes sieve, we have identified discrete dynamic systems that exactly model the populations of gaps across stages of Eratosthenes sieve. Our models provide some insight into the observed biases in the occurrences of last digits in consecutive primes, and the models suggest that the biases will ultimately be reversed for large enough primes. The exact model for populations of gaps across stages of Eratosthenes…
