Related papers: Bounded gaps between primes in number fields and f…
The Hardy-Littlewood method is a well-known technique in analytic number theory. Among its spectacular applications are Vinogradov's 1937 result that every sufficiently large odd number is a sum of three primes, and a related result of…
We show that the existence of arithmetic progressions with few primes, with a quantitative bound on "few", implies the existence of larger gaps between primes less than x than is currently known unconditionally. In particular, we derive…
The article presents a generalization of the classical Hardy-Littlewood conjecture concerning the density of prime tuples to the case of tuples consisting of almost-prime numbers (numbers with a specified quantity of prime divisors). The…
In the present work the existence of some patterns of primes is shown which generalize the celebrated result of Green and Tao according to which there are arbitrarily long arithmetic progressions in the sequence of primes
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…
In 1922 Hardy and Littlewood proposed a conjecture on the asymptotic density of admissible prime k-tuples. In 2011 Wolf computed the "Skewes number" for twin primes, i.e., the first prime at which a reversal of the Hardy-Littlewood…
A long-standing conjecture of Littlewood about simultaneous Diophantine approximation has an analogous problem for a field of formal Laurent series $\mathbb{F}(\!(t^{-1})\!)$. That is, we can ask whether for any series $\Theta$, $\Phi$ and…
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…
We prove a number field analogue of the Green--Tao--Ziegler theorem on simultaneous prime values of degree 1 polynomials whose linear parts are pairwise linearly independent. Applications of our results include a Hasse principle of rational…
Let $\k$ be an arbitrary field. For any fixed badly approximable power series $\Theta$ in $\k((X^{-1}))$, we give an explicit construction of continuum many badly approximable power series $\Phi$ for which the pair $(\Theta, \Phi)$…
While the sequence of primes is very well distributed in the reduced residue classes (mod $q$), the distribution of pairs of consecutive primes among the permissible $\phi(q)^2$ pairs of reduced residue classes (mod $q$) is surprisingly…
We show that for infinitely many primes $p$, there exist dual functions of order $k$ over $\mathbb{F}_p^n$ that cannot be approximated in $L_\infty$-distance by polynomial phase functions of degree $k-1$. This answers in the negative a…
We prove that the primes of the form $x^2+y^2+1$ contain arbitrarily long non-trivial arithmetic progressions.
The aim of this paper is to try to establish a generic model for the problem that several multivariable number-theoretic functions represent simultaneously primes for infinitely many integral points. More concretely, we introduced briefly…
This is an expository article to accompany my two lectures at the CDM conference. I have used this an excuse to make public two sets of notes I had lying around, and also to put together a short reader's guide to some recent joint work with…
As a corollary to the recent extraordinary theorem of Maynard and Tao, we re-prove, in a stronger form, a result of Shiu concerning "strings" of consecutive, congruent primes.
We prove an analogue of a result by Goldston, Pintz and Yildirim for small gaps between primes that split completely in an abelian number field. We prove both a conditional result assuming the Elliott-Halberstam conjecture, and an…
For any integer $d \geq 2$ and prime power $q$, we construct unexpectedly large induced matchings in the point-line incidence graph of $\mathbb{F}_{q}^{d}$ by leveraging a new connection with the Furstenberg-S\'ark\"ozy problem from…
We prove that there are infinitely often pairs of primes much closer than the average spacing between primes - almost within the square root of the average spacing. We actually prove a more general result concerning the set of values taken…
There is extensive numerical support for the prime-pair conjecture (PPC) of Hardy and Littlewood (1923) on the asymptotic behavior of pi_{2r}(x), the number of prime pairs (p,p+2r) with p not exceeding x. However, it is still not known…