Related papers: The mixed Littlewood conjecture for pseudo-absolut…

Let $\zeta = \xi + i\xi'$ where $\xi, \xi'$ are iid copies of a mean zero, variance one, subgaussian random variable. Let $N_n$ be a $n \times n$ random matrix with entries that are iid copies of $\zeta$. We prove that there exists a $c \in…

The Sendov conjecture asserts that if all the zeros of a polynomial p lie in the closed unit disk then there must be a zero of p ' within unit distance of each zero. In this paper we give a partial result when p has simple zeros.

It has been known since Erdos that the sum of $1/(n\log n)$ over numbers $n$ with exactly $k$ prime factors (with repetition) is bounded as $k$ varies. We prove that as $k$ tends to infinity, this sum tends to 1. Banks and Martin have…

We consider the probability that the random signed sum $\xi_1 v_1 + \dotsb + \xi_n v_n$ lies within a given distance $r$ of the origin, where $v_1,\dotsc,v_n \in \mathbb{R}^d$ are fixed unit vectors and $\xi_1,\dotsc,\xi_n$ are…

A symmetric subset of the reals is one that remains invariant under some reflection z --> c-z. We consider, for any 0 < x <= 1, the largest real number D(x) such that every subset of $[0,1]$ with measure greater than x contains a symmetric…

We prove an intermediate value theorem of an arithmetical flavor, involving the consecutive averages of sequences with terms in a given finite set A. For every such set we completely characterize the numbers x ("intermediate values") with…

Cilleruelo conjectured that if $f\in\mathbb{Z}[x]$ is an irreducible polynomial of degree $d\ge 2$ then, $\log \operatorname{lcm} \{f(n)\mid n<x\} \sim (d-1)x\log x.$ In this article, we investigate the analogue of prime arguments, namely,…

Partial rigidity is a quantitative notion of recurrence and provides a global obstruction which prevents the system from being strongly mixing. A dynamical system $(X, \mathcal{X}, \mu, T)$ is partially rigid if there is a constant $\delta…

The well-known Leibniz theorem (Leibniz Criterion or alternating series test) of convergence of alternating series is generalized for the case when the absolute value of terms of series are "not absolutely monotonously" convergent to zero.…

Assuming the Riemann hypothesis, this article discusses a new elementary argument that seems to prove that the maximal prime gap of a finite sequence of primes p_1, p_2, ..., p_n <= x, satisfies max {p_(n+1) - p_n : p_n <= x} <=…

We address the question of the infinitude of twin and cousin prime pairs from a probabilistic perspective. Our approach partitions the set of integer numbers greater than $2$ in finite intervals of the form $[p_{n-1}^2,p_n^2)$, $p_{n-1}$…

We prove a conjecture of B. Gr\"unbaum stating that the set of affine invariant points of a convex body equals to the set of points invariant under all affine linear symmetries of the convex body. As a consequence we give a short proof on…

For any given real number $\alpha$ with bounded partial quotients, we construct explicitly continuum many real numbers $\beta$ with bounded partial quotients for which the pair $(\alpha, \beta)$ satisfies a strong form of the Littlewood…

In a previous paper, we studied certain sequences of simultaneous rational approximations in ${\bf R}^2$ which present some analogy with the continued fractions. We got results around the Littlewood conjecture by using such approximations.…

We determine a necessary and sufficient condition for the infinitude of primes $p$ such that none of the equations $a_i^x \equiv b_i \pmod{p}, 1 \le i \le n,$ are solvable. We control the insolvability of $a^x \equiv b \pmod{p}$ by power…

We prove the exceptional zero conjecture for the symmetric powers of CM cuspidal eigenforms at ordinary primes. In other words, we determine the trivial zeroes of the associated p-adic L-functions, compute the L-invariants, and show that…

Pick $n$ independent and uniform random points $U_1,\ldots,U_n$ in a compact convex set $K$ of $\mathbb{R}^d$ with volume 1, and let $P^{(d)}_K(n)$ be the probability that these points are in convex position. The Sylvester conjecture in…

Let $\chi$ be a real primitive character to the modulus $D$. It is proved that $$ L(1,\chi)\gg (\log D)^{-2022} $$ where the implied constant is absolute and effectively computable. In the proof, the lower bound for $L(1,\chi)$ is first…

We introduce a wide class of deterministic subsets of primes of zero relative density and we prove Roth's Theorem in these sets, namely, we show that any subset of them with positive relative upper density contains infinitely many…

It is the purpose of this thesis to enunciate and prove a collection of explicit results in the theory of prime numbers. First, the problem of primes in short intervals is considered. We prove that there is a prime between consecutive cubes…