Related papers: On the ratio of consecutive gaps between primes

Let $(X,B_X,\mu,T)$ be a measure-preserving probability system with $T$ is invertible. Suppose that $A\in B_X$ with $\mu(A)>0$ and $\epsilon>0$. For any $m\geq 1$, there exist infinitely many primes $p_0,p_1,\ldots,p_m$ with…

We examine the prime gaps using a statistical approach. It is first shown that the Andrica's conjecture is true for half or more cases. Using the arguments of averages, it is further shown that Andrica's conjecture is true. We further…

For $j=1,2$, let $f_j(z) = \sum_{n=1}^{\infty} a_{j}(n) e^{2\pi i nz}$ be a holomorphic, non-CM cuspidal newform of even weight $k_j \ge 2$ with trivial nebentypus. For each prime $p$, let $\theta_{j}(p)\in[0,\pi]$ be the angle such that…

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…

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…

According to a classical result of Spencer, Szemer\'edi, and Trotter (1984), the maximum number of times the unit distance can occur among $n$ points in the plane is $O(n^{4/3})$. This is far from Erd\H{o}s's lower bound, $n^{1+O(1/\log\log…

This paper updates the explicit interval estimate for primes between consecutive powers. It is shown that there is least one prime between $n^{155}$ and $(n+1)^{155}$ for all $n\geq 1$. This result is in part obtained with a new explicit…

We investigate the behaviour of a certain additive function depending on prime divisors of specific integers lying in large gaps between consecutive primes. The result is obtained by a combination of results and ideas related to large gaps…

Let $p_n$ denote the $n$-th prime. For any $m\geq 1$, there exist infinitely many $n$ such that $p_{n}-p_{n-m}\leq C_m$ for some large constant $C_m>0$, and $$p_{n+1}-p_n\geq \frac{c_m\log n\log\log n\log\log\log\log n}{\log\log\log n}, $$…

In a previous paper, the authors proved that in any system of three linear forms satisfying obvious necessary local conditions, there are at least two forms that infinitely often assume $E_2$-values; i.e., values that are products of…

We give an exposition of some connections between Fourier optimization problems and problems in number theory. In particular, we present some recent conditional bounds under the generalized Riemann hypothesis, achieved via a Fourier…

In the present work a new simple proof of the theorem of Gallagher about the average of the singular series in the Hardy-Littlewood prime k-tuple conjecture is proved (in an even stronger form) which is uniform with respect to k (if the…

Let $m \in \mathbb{N}$ be large. We show that there exist infinitely many primes $q_{1}< \cdot\cdot\cdot < q_{m+1}$ such that \[ q_{m+1}-q_{1}=O(e^{7.63m}) \] and $q_{j}+2$ has at most \[ \frac{7.36m}{\log 2} + \frac{4\log m}{\log 2} + 21…

The set of short intervals between consecutive primes squared has the pleasant---but seemingly unexploited---property that each interval $s_k:=\{p_k^2, \dots,p_{k+1}^2-1\}$ is fully sieved by the $k$ first primes. Here we take advantage of…

We study the gaps between products of two primes in imaginary quadratic number fields using a combination of the methods of Goldston-Graham-Pintz-Yildirim \cite{GGPY}, and Maynard \cite{MAY}. An important consequence of our main theorem is…

This paper introduces a new method to find the next prime number after a given prime ${P}$. The proposed method is used to derive a system of inequalities, that serve as constraints which should be satisfied by all primes whose successor is…

The celebrated Green-Tao theorem states that the prime numbers contain arbitrarily long arithmetic progressions. We give an exposition of the proof, incorporating several simplifications that have been discovered since the original paper.

We prove a couple of related theorems including Legendre's and Andrica's conjecture. Key to the proofs is an algorithm that delivers the exact upper bound on the greatest gap that can occur in a combinatorial game with the set of P primes…

We discuss recent advances on weak forms of the Prime $k$-tuple Conjecture, and its role in proving new estimates for the existence of small gaps between primes and the existence of large gaps between primes.

We present effective upper bounds on the symmetric bilinear complexity of multiplication in extensions of a base finite field Fp2 of prime square order, obtained by combining estimates on gaps between prime numbers together with an optimal…