Related papers: A new bound on the relative error in the Chebotare…

For Artin-Schreier curve y^q -y = f(x) defined over a finite field F_q of q elements, we show that the Weil bound for the number of the rational points over extension fields of F_q can often be greatly improved, essentially removing an…

This paper studies explicit and theoretical bounds for several interesting quantities in number theory, conditionally on the Generalized Riemann Hypothesis. Specifically, we improve the existing explicit bounds for the least quadratic…

We explain why we hope that the Froissart bound can be improved, either qualitatively or, more likely, quantitatively, by making a better use of unitarity, in particular elastic unitarity. In other instances (Gribov's theorem) elastic…

We show how to use the Malliavin calculus to obtain density estimates of the law of general centered random variables. In particular, under a non-degeneracy condition, we prove and use a new formula for the density of a random variable…

We generalize the classical Bombieri-Vinogradov theorem to a short interval, non-abelian setting. This leads to variants of the prime number theorem for short intervals where the primes lie in arithmetic progressions that are "twisted" by a…

We realize infinitely many covering groups $2.A_n$ (where $A_n$ is the alternating group) as the Galois group of everywhere unramified Galois extensions over infinitely many quadratic number fields. After several predecessor works…

We establish an improved form of the classical logarithmic Sobolev inequality for the Gaussian measure restricted to probability densities which satisfy a Poincar\'e inequality. The result implies a lower bound on the deficit in terms of…

Let $n \geq 6$ be an integer. We prove that the number of number fields with Galois group $A_n$ and absolute discriminant at most $X$ is asymptotically at least $X^{1/8 + O(1/n)}$. For $n \geq 8$ this improves upon the previously best known…

Extending the works of Alladi and Sweeting and Woo, we state and prove the general higher order duality between prime ideals in number rings. We then use the second order duality to obtain the a new formula for the Chebotarev Density…

Let $Z \to X$ be a finite branched Galois cover of normal projective geometrically integral varieties of dimension $d \geq 2$ over a perfect field $k$. For such a cover, we prove a Chebotarev-type density result describing the decomposition…

Let $K$ be a number field and $S$ a set of primes of $K$. We write $K_S/K$ for the maximal extension of $K$ unramified outside $S$ and $G_{K,S}$ for its Galois group. In this paper, we prove the following generalization of the…

Let L/k be a finite Galois extension of number fields with Galois group G. For every odd prime p satisfying certain mild technical hypotheses, we use values of Artin L-functions to construct an element in the centre of the group ring…

We generalise Gabidulin codes to the case of infinite fields, eventually with characteristic zero. For this purpose, we consider an abstract field extension and any automorphism in the Galois group. We derive some conditions on the…

We confirm Chebyshev's observation that primes are strikingly more abundant in non-square residue classes modulo a fixed integer under the Generalized Riemann Hypothesis (GRH) by proving a (natural) density $1$ statement for prime counting…

This paper re-examines the limit theorems of Abadie and Imbens for nearest-neighbor matching estimators of average treatment effects with a fixed number of matches. We establish, for the first time, a non-normalized central limit theorem…

We establish a joint distribution result concerning the fractional part of $\alpha p^\theta$ for $\theta \in (0,1), \ \alpha>0$, where $p$ is a prime satisfying a Chebotarev condition in a fixed finite Galois extension over $\mathbb{Q}$. As…

We consider the problem of linear estimation, and establish an extension of the Gauss-Markov theorem, in which the bias operator is allowed to be non-zero but bounded with respect to a matrix norm of Schatten type. We derive simple and…

Let A be an abelian variety defined over a number field and of dimension g. When g<3, by the recent work of Sawin, we know the exact (nonzero) value of the density of the set of primes which are ordinary for A. In higher dimension very…

Suppose given a Galois etale cover Y -> X of proper non-singular curves over an algebraically closed field k of prime characteristic p. Let H be its Galois group, and G any finite extension of H by a p-group P. We give necessary and…

We show that there exists $c>0$ such that any subset of $\{1, \dots, N\}$ of density at least $(\log\log{N})^{-c}$ contains a nontrivial progression of the form $x,x+y,x+y^2$. This is the first quantitatively effective version of the…