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 questions posed by Erdos and Turan. The dynamic system underlying the recursion provides evidence toward Polignac's conjecture and in support of the estimates made for gaps among primes by Hardy and Littlewood in Conjecture B of their 1923 paper.
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.
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. 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…
We show how to carry out a sieve of Eratosthenes up to N in space O(N^{1/3} (log N)^{2/3}) and time O(N log N). These bounds constitute an improvement over the usual versions of the sieve, which take space about O(sqrt{N}) and time at least linear on N. We can also apply our sieve to any subinterval of [1,N] of length Omega(N^{1/3} (log N)^{2/3}) in time essentially linear on the length of the interval. Before, such a thing was possible only for subintervals of [1,N] of length Omega(sqrt{N}). Just as in (Galway, 2000), our approach is related to Diophantine approximation, and also has close ti…
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.
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 are known as constellations. As the recursion proceeds, adjacent gaps within longer constellations are added together to produce shorter constellations of the same sum. These additions or closures correspond to removing composite numbers that are divisible by the prime for that stage of Eratosthenes sieve. Although we don't know where in the cycle of gaps a closure wil…
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…
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…
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 are known as constellations. As the recursion proceeds, adjacent gaps within longer constellations are added together to produce shorter constellations of the same sum. These additions or closures correspond to removing composite numbers that are divisible by the prime for that stage of Eratosthenes sieve. Although we don't know where in the cycle of gaps a closure wil…
An arithmetic function $f$ is called a $sieve$ $function$ of $range$ $Q$ if its Eratosthenes transform $g=f\astμ$ has support in $[1,Q]$, where $g(q)\ll_{\varepsilon} q^{\varepsilon}$ ($\forall\varepsilon>0$). We continue our study of the distribution of such functions over short $arithmetic$ $bands$, $n\equiv ar+b\, (\bmod\,q)$, with $1\le a\le H=o(N)$ and $r,b$ integers such that g.c.d.$(r,q)=1$. In particular, we discuss the optimality of some results.
An arithmetic function $f$ is called a {\it sieve function of range} $Q$, if its Eratosthenes transform $g=f\astμ$ is supported in $[1,Q]\cap\N$, where $g(q)\ll_{\varepsilon} q^{\varepsilon}$ ($\forall\varepsilon>0$). Here, we study the distribution of $f$ over short {\it arithmetic bands} $\cup_{1\le a\le H}\{n\in(N,2N]: n\equiv a\, (\bmod\,q)\}$, with $H=o(N)$, and give applications to both the correlations and to the so-called weighted Selberg integrals of $f$, on which we have concentrated our recent research.
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…
We establish an asymptotic formula for the number of k-difference twin primes associated with the Hawkins random sieve, which is a probabilistic model of the Eratosthenes sieve. The formula for k=1 was obtained by Wunderlich [Acta Arith. 26 (1974), 59--81]. We here extend this to k \geq 2 and generalize it to all l-tuples of Hawkins primes.
We study the gaps between consecutive prime numbers directly through Eratosthenes sieve. Using elementary methods, we identify a recursive relation for these gaps and for specific sequences of consecutive gaps, known as constellations. Using this recursion we can estimate the numbers of a gap or of a constellation that occur between a prime and its square. This recursion also has explicit implications for open questions about gaps between prime numbers, including three questions posed by Erdös and Turán.
One field of particular interest in Number Theory concerns the gaps between consecutive primes. Within the last few years, very important results have been achieved on how small these gaps can be. The strongest of these results were obtained by Dan Goldston, Janos Pintz and Cem Yalcin Yildirim. The present work begins by generalizing their results so that they can be applied to related problems in a more direct manner. Additionally, we improve the bound for $F_2$ (concerning the maximal gap in a block of three primes) obtained by the authors' first paper with our generalization.
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…
Using Duke's large sieve inequality for Hecke Gr{ö}ssencharaktere and the new sieve methods of Maynard and Tao, we prove a general result on gaps between primes in the context of multidimensional Hecke equidistribution. As an application, for any fixed $0<ε<\frac{1}{2}$, we prove the existence of infinitely many bounded gaps between primes of the form $p=a^2+b^2$ such that $|a|<ε\sqrt{p}$. Furthermore, for certain diagonal curves $\mathcal{C}:ax^α+by^β=c$, we obtain infinitely many bounded gaps between the primes $p$ such that $|p+1-\#\mathcal{C}(\mathbb{F}_p)|<ε\sqrt{p}$.
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.
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 are known as constellations. In this paper, we use those enumerations to estimate the numbers of these constellations that occur as constellations among prime numbers, and we compare these estimates with computational results. We include in our estimates the constellations corresponding to three and four consecutive primes in arithmetic progression. For these initial e…
Let $r \ge 2$ be an integer. We adapt the Maynard-Tao sieve to produce the asymptotically best-known bounded gaps between products of $r$ distinct primes. Our result applies to positive-density subsets of the primes that satisfy certain equidistribution conditions. This improves on the work of Thorne and Sono.
We show that there exists pairs of consecutive primes less than $x$ whose difference is larger than $t(1+o(1))(\log{x})(\log\log{x})(\log\log\log\log{x})(\log\log\log{x})^{-2}$ for any fixed $t$. Our proof works by incorporating recent progress in sieve methods related to small gaps between primes into the Erdos-Rankin construction. This answers a well-known question of Erdos.
We use the asymptotic large sieve, developed by the authors, to prove that if the Generalized Riemann Hypothesis is true, then there exist many Dirichlet L-functions that have a pair of consecutive zeros closer together than 0.37 times their average spacing. More generally, we investigate zero spacings within the family of twists by Dirichlet characters of a fixed L-function and give precise bounds for small gaps which depend only on the degree of the L-function.
We introduce a refinement of the GPY sieve method for studying prime $k$-tuples and small gaps between primes. This refinement avoids previous limitations of the method, and allows us to show that for each $k$, the prime $k$-tuples conjecture holds for a positive proportion of admissible $k$-tuples. In particular, $\liminf_{n}(p_{n+m}-p_n)<\infty$ for any integer $m$. We also show that $\liminf(p_{n+1}-p_n)\le 600$, and, if we assume the Elliott-Halberstam conjecture, that $\liminf_n(p_{n+1}-p_n)\le 12$ and $\liminf_n (p_{n+2}-p_n)\le 600$.
We show that at least 1/3 of positive real numbers are in the set of limit points of normalized prime gaps. More precisely, if $p_n$ denotes the $n$th prime and $\mathbb{L}$ is the set of limit points of the sequence $\{(p_{n+1}-p_n)/\log p_n\}_{n=1}^\infty,$ then for all $T\geq 0$ the Lebesque measure of $\mathbb{L} \cap [0,T]$ is at least $T/3.$ This improves the result of Pintz (2015) that the Lebesque measure of $\mathbb{L} \cap [0,T]$ is at least $(1/4-o(1))T,$ which was obtained by a refinement of the previous ideas of Banks, Freiberg, and Maynard (2015). Our improvement comes from using…
In the present work we prove a number of surprising results about gaps between consecutive primes and arithmetic progressions in the sequence of generalized twin primes which could not have been proven without the recent fantastic achievement of Yitang Zhang about the existence of bounded gaps between consecutive primes. Most of these results would have belonged to the category of science fiction a decade ago. However, the presented results are far from being immediate consequences of Zhang's famous theorem: they require various new ideas, other important properties of the applied sieve functi…
