Related papers: Large prime gaps and probabilistic models

We generalize current known distribution results on Shanks--R\'enyi prime number races to the case where arbitrarily many residue classes are involved. Our method handles both the classical case that goes back to Chebyshev and function…

Dickson conjectured that a set of polynomials will take on infinitely many simultaneous prime values. Later others, such as Hardy and Littlewood, gave estimates for the number of these primes. In this article we look at this conjecture,…

Upper bounds on the maximum number of codewords in a binary code of a given length and minimum Hamming distance are considered. New bounds are derived by a combination of linear programming and counting arguments. Some of these bounds…

We describe a generalization of the large sieve to situations where the underlying groups are nonabelian, and give several applications to the arithmetic of abelian varieties. In our applications, we sieve the set of primes via the system…

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…

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…

Recent advances in statistical inference have significantly expanded the toolbox of probabilistic modeling. Historically, probabilistic modeling has been constrained to (i) very restricted model classes where exact or approximate…

We implement the Maynard-Tao method of detecting primes in tuples to investigate small gaps between primes in arithmetic progressions, with bounds that are uniform over a range of moduli.

For positive integers $q$, Dirichlet's theorem states that there are infinitely many primes in each reduced residue class modulo $q$. A stronger form of the theorem states that the primes are equidistributed among the $\varphi(q)$ reduced…

We give a finite-sample analysis of predictive inference procedures after model selection in regression with random design. The analysis is focused on a statistically challenging scenario where the number of potentially important…

We generalise Zhang's and Pintz recent results on bounded prime gaps to give a lower bound for the the number of prime pairs bounded by 6*10^7 in the short interval $[x,x+x (\log x)^{-A}]$. Our result follows only by analysing Zhang's proof…

We consider the problem of variable selection in linear models when $p$, the number of potential regressors, may exceed (and perhaps substantially) the sample size $n$ (which is possibly small).

The prime counting function inequality $\pi(x+y) < \pi(x)+\pi(y)$, which is known as Hardy-Littlewood conjecture, has been established for a variety of cases such as $ \delta x \leq y \leq x$, where $0< \delta \leq 1$, and $x \leq y\leq x…

Let $f$ be a primitive positive definite integral binary quadratic form of discriminant $-D$ and let $\pi_f(x)$ be the number of primes up to $x$ which are represented by $f$. We prove several types of upper bounds for $\pi_f(x)$ within a…

In the framework of Cramer's probabilistic model of primes, we explore the exact and asymptotic distributions of maximal prime gaps. We show that the Gumbel extreme value distribution exp(-exp(-x)) is the limit law for maximal gaps between…

In this article we consider the problem of choosing an optimal sampling scheme for the regression problem simultaneously with that of model selection. We consider a batch type approach and an on-line approach following algorithms recently…

We present a new technique for proving logarithmic upper bounds for diameters of evolving random graph models, which is based on defining a coupling between random graphs and variants of random recursive trees. The advantage of the…

The purpose of this paper is twofold: 1) Applications of Gallagher's larger sieve modulo prime squares do not work. In some relevant cases we can transform the residue class information modulo $p^2$ to more suitable residue information…

Let $\pi(x;q,a)$ denote the number of primes up to $x$ that are congruent to $a$ modulo $q$. A prime number race, for fixed modulus $q$ and residue classes $a_1, \ldots, a_r$, investigates the system of inequalities $\pi(x;q,a_1) >…

Building on previous results of Xing, we give new lower bounds on the rate of intersecting codes over large alphabets. The proof is constructive, and uses algebraic geometry, although nothing beyond the basic theory of linear systems on…