Jacques Benatar

activity

20172024

collaborators

7 papers

math.NT2024

A short-interval Hildebrand-Tenenbaum theorem

Jacques Benatar

In the late eighties, Hildebrand and Tenenbaum proved an asymptotic formula for the number of positive integers below $x$ , having exactly $ν$ distinct prime divisors: $π_ν(x) \sim…

math.NT2022

Extremal bounds for Dirichlet polynomials with random multiplicative coefficients

Jacques Benatar, Alon Nishry

For $X (n)$ a Steinhaus random multiplicative function, we study the maximal size of the random Dirichlet polynomial $D_{N} (t) = \frac{1}{N} n \leq N \sum X (n) n^{i t},$ wit…

math.CV2021

Zero distribution of power series and binary correlation of coefficients

Jacques Benatar, Alexander Borichev, Mikhail Sodin

We study the distribution of zeroes of power series with infinite radius of convergence. The coefficients of the series have the form $ξ (n) a (n)$ , where $a$ is a smooth sequence of…

math.NT2020

Moments of polynomials with random multiplicative coefficients

Jacques Benatar, Alon Nishry, Brad Rodgers

For $X (n)$ a Rademacher or Steinhaus random multiplicative function, we consider the random polynomials $P_{N} (θ) = \frac{1}{N} n \leq N \sum X (n) e (n θ),$ and show that the…

math.PR2019

The "pits effect" for entire functions of exponential type and the Wiener spectrum

Jacques Benatar, Alexander Borichev, Mikhail Sodin

Given a sequence $ξ : Z_{+} \to C$ , we find a simple spectral condition which guarantees the angular equidistribution of the zeroes of the Taylor series \[ F_ξ(z)…

math-ph2017

Planck-scale distribution of nodal length of arithmetic random waves

Jacques Benatar, Domenico Marinucci, Igor Wigman

We study the nodal length of random toral Laplace eigenfunctions ("arithmetic random waves") restricted to decreasing domains ("shrinking balls"), all the way down to Planck scale.…