An official website of the United States government
Abstarct Given disjoint subsets T 1 , …, T m of “not too large” primes up to x , we establish that for a random integer n drawn from [1, x ], the m -dimensional vector enumerating the number of prime factors of n from T 1 , …, T m converges to a vector of m independent Poisson random variables. We give a specific rate of convergence using the Kubilius model of prime factors. We also show a universal upper bound of Poisson type when T 1 , …, T m are unrestricted, and apply this to the distribution of the number of prime factors from a set T conditional on n having k total prime factors.
more »
« less
Large prime gaps and probabilistic models
Abstract We introduce a new probabilistic model of the primes consisting of integers that survive the sieving process when a random residue class is selected for every prime modulus below a specific bound. From a rigorous analysis of this model, we obtain heuristic upper and lower bounds for the size of the largest prime gap in the interval $[1,x]$ [ 1 , x ] . Our results are stated in terms of the extremal bounds in the interval sieve problem. The same methods also allow us to rigorously relate the validity of the Hardy-Littlewood conjectures for an arbitrary set (such as the actual primes) to lower bounds for the largest gaps within that set.
more »
« less
- PAR ID:
- 10440641
- Date Published:
- Journal Name:
- Inventiones mathematicae
- Volume:
- 233
- Issue:
- 3
- ISSN:
- 0020-9910
- Page Range / eLocation ID:
- 1471 to 1518
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Abstract We develop an analog for shifted primes of the Kubilius model of prime factors of integers. We prove a total variation distance estimate for the difference between the model and actual prime factors of shifted primes, and apply it to show that the prime factors of shifted primes in disjoint sets behave like independent Poisson variables. As a consequence, we establish a transference principle between the anatomy of random integers $$\leqslant x$$ and of random shifted primes p+a with $$p\leqslant x$$.more » « less
-
null (Ed.)In this paper, we make partial progress on a function field version of the dynamical uniform boundedness conjecture for certain one-dimensional families $${\mathcal{F}}$$ of polynomial maps, such as the family $$f_{c}(x)=x^{m}+c$$ , where $$m\geq 2$$ . We do this by making use of the dynatomic modular curves $$Y_{1}(n)$$ (respectively $$Y_{0}(n)$$ ) which parametrize maps $$f$$ in $${\mathcal{F}}$$ together with a point (respectively orbit) of period $$n$$ for $$f$$ . The key point in our strategy is to study the set of primes $$p$$ for which the reduction of $$Y_{1}(n)$$ modulo $$p$$ fails to be smooth or irreducible. Morton gave an algorithm to construct, for each $$n$$ , a discriminant $$D_{n}$$ whose list of prime factors contains all the primes of bad reduction for $$Y_{1}(n)$$ . In this paper, we refine and strengthen Morton’s results. Specifically, we exhibit two criteria on a prime $$p$$ dividing $$D_{n}$$ : one guarantees that $$p$$ is in fact a prime of bad reduction for $$Y_{1}(n)$$ , yet this same criterion implies that $$Y_{0}(n)$$ is geometrically irreducible. The other guarantees that the reduction of $$Y_{1}(n)$$ modulo $$p$$ is actually smooth. As an application of the second criterion, we extend results of Morton, Flynn, Poonen, Schaefer, and Stoll by giving new examples of good reduction of $$Y_{1}(n)$$ for several primes dividing $$D_{n}$$ when $n=7,8,11$ , and $$f_{c}(x)=x^{2}+c$$ . The proofs involve a blend of arithmetic and complex dynamics, reduction theory for curves, ramification theory, and the combinatorics of the Mandelbrot set.more » « less
-
Abstract We present efficient algorithms for counting points on a smooth plane quartic curve X modulo a prime p . We address both the case where X is defined over $${\mathbb {F}}_p$$ F p and the case where X is defined over $${\mathbb {Q}}$$ Q and p is a prime of good reduction. We consider two approaches for computing $$\#X({\mathbb {F}}_p)$$ # X ( F p ) , one which runs in $$O(p\log p\log \log p)$$ O ( p log p log log p ) time using $$O(\log p)$$ O ( log p ) space and one which runs in $$O(p^{1/2}\log ^2p)$$ O ( p 1 / 2 log 2 p ) time using $$O(p^{1/2}\log p)$$ O ( p 1 / 2 log p ) space. Both approaches yield algorithms that are faster in practice than existing methods. We also present average polynomial-time algorithms for $$X/{\mathbb {Q}}$$ X / Q that compute $$\#X({\mathbb {F}}_p)$$ # X ( F p ) for good primes $$p\leqslant N$$ p ⩽ N in $$O(N\log ^3 N)$$ O ( N log 3 N ) time using O ( N ) space. These are the first practical implementations of average polynomial-time algorithms for curves that are not cyclic covers of $${\mathbb {P}}^1$$ P 1 , which in combination with previous results addresses all curves of genus $$g\leqslant 3$$ g ⩽ 3 . Our algorithms also compute Cartier–Manin/Hasse–Witt matrices that may be of independent interest.more » « less
-
We show that, for almost all x x , the interval ( x , x + ( log x ) 2.1 ] (x, x+(\log x)^{2.1}] contains products of exactly two primes. This improves on a work of the second author that had 3.51 3.51 in place of 2.1 2.1 . To obtain this improvement, we prove a new type II estimate. One of the new innovations is to use Heath-Brown’s mean value theorem for sparse Dirichlet polynomials.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s00222-023-01199-0
