49 citations · 178 across the 28 of their papers we have counts for
37 papers
A lower bound for the number of Egyptian fractions
Sandro Bettin, Loïc Grenié, Giuseppe Molteni +1
An Egyptian fraction is a sum of the form where are distinct positive integers. We prove explicit lower bounds for the cardinality of the…
On products of sets of natural density one
Sandro Bettin, Matteo Bordignon, Alessandro Fazzari
In a previous work, Bettin, Koukoulopoulos, and Sanna prove that if two sets of natural numbers and have natural density , then their product set $A \cdot B := \{ab : a…
On quantum modular forms of non-zero weights
Sandro Bettin, Sary Drappeau
We study functions on which statisfy a ``quantum modularity'' relation of the shape where $h:\mathbb R_{\ne…
A weighted one-level density of the non-trivial zeros of the Riemann zeta-function
Sandro Bettin, Alessandro Fazzari
We compute the one-level density of the non-trivial zeros of the Riemann zeta-function weighted by for and, for test functions with Fourier support in…
Ranks of elliptic curves over of small degree in
Francesco Battistoni, Sandro Bettin, Christophe Delaunay
We study elliptic surfaces over with coefficients of a Weierstrass model being polynomials in with degree at most 2. We derive an explicit expressio…
On the typical rank of elliptic curves over
Francesco Battistoni, Sandro Bettin, Christophe Delaunay
We give upper bounds for the number of rational elliptic surfaces in some families having positive rank, obtaining in particular that these form a subset of density zero. This conf…
Effective estimation of some oscillatory integrals related to infinitely divisible distributions
Sandro Bettin, Sary Drappeau
We present a practical framework to prove, in a simple way, two-terms asymptotic expansions for Fourier integrals $$ {\mathcal I}(t) = \int_{\mathbb R}({\rm e}^{itϕ(x)}-1) {\rm d}…
A note on the natural density of product sets
Sandro Bettin, Dimitris Koukoulopoulos, Carlo Sanna
Given two sets of natural numbers and of natural density we prove that their product set $\mathcal{A}\cdot \mathcal{B}:=\{ab:a\in\mathcal{A},\,b\in\…
Two arithmetic applications of perturbations of composition operators
Sandro Bettin, Sary Drappeau
We estimate the spectral radius of perturbations of a particular family of composition operators, in a setting where the usual choices of norms do not account for the typical size…
Averages of long Dirichlet polynomials
Sandro Bettin, J. Brian Conrey
We consider the asymptotic behavior of the mean square of truncations of the Dirichlet series of . We discuss the connections of this problem with that of the variance of t…
Counting Egyptian fractions
Sandro Bettin, Loïc Grenié, Giuseppe Molteni +1
For any integer , let be the set of all Egyptian fractions employing denominators less than or equal to . We give upper and lower bounds for the cardi…
Modularity and value distribution of quantum invariants of hyperbolic knots
Sandro Bettin, Sary Drappeau
We obtain an exact modularity relation for the -Pochhammer symbol. Using this formula, we show that Zagier's modularity conjecture for a knot essentially reduces to the arit…
Partial sums of the cotangent function
Sandro Bettin, Sary Drappeau
Nous prouvons l'existence de formules de réciprocité pour des sommes de la forme , où est une fonction par morceaux, qui…
Limit laws for rational continued fractions and value distribution of quantum modular forms
Sandro Bettin, Sary Drappeau
We study the limiting distributions of Birkhoff sums of a large class of cost functions (observables) evaluated along orbits, under the Gauss map, of rational numbers in or…
Mixed moments of characteristic polynomials of random unitary matrices
Emma C. Bailey, Sandro Bettin, Gordon Blower +4
Following the work of Conrey, Rubinstein and Snaith and Forrester and Witte we examine a mixed moment of the characteristic polynomial and its derivative for matrices from the unit…
A note on the dimension of the largest simple Hecke submodule
Sandro Bettin, Corentin Perret-Gentil, Maksym Radziwiłł
For even, let denote the dimension of the largest simple Hecke submodule of . We show, using a simple analytic method, that…
Small values of signed harmonic sums
Sandro Bettin, Giuseppe Molteni, Carlo Sanna
For every and every integer , let be the minimum of the distance of from the sums , where $s_1, \ldots, s_n \in \{-1…
The power-saving Manin-Peyre's conjectures for a senary cubic
Sandro Bettin, Kevin Destagnol
Using recent work of the first author~\cite{Bet}, we prove a strong version of the Manin-Peyre's conjectures with a full asymptotic and a power-saving error term for the two variet…
Greedy approximations by signed harmonic sums and the Thue--Morse sequence
Sandro Bettin, Giuseppe Molteni, Carlo Sanna
Given a real number , we study the approximation of by signed harmonic sums , where the sequence of signs i…
On the non-vanishing of certain Dirichlet series
Sandro Bettin, Bruno Martin
Given , we study the vanishing of the Dirichlet series at the point , where is a periodic function modulo a pri…
Statistical distribution of the Stern sequence
Sandro Bettin, Sary Drappeau, Lukas Spiegelhofer
We prove that the Stern diatomic sequence is asymptotically distributed according to a normal law, on a logarithmic scale. This is obtained by studying complex moments, and the ana…
A conjectural extension of Hecke's converse theorem
Sandro Bettin, Jonathan W. Bober, Andrew R. Booker +6
We formulate a precise conjecture that, if true, extends the converse theorem of Hecke without requiring hypotheses on twists by Dirichlet characters or an Euler product. The main…
Linear correlations of the divisor function
Sandro Bettin
Motivated by arithmetic applications on the number of points in a bihomogeneous variety and on moments of Dirichlet -functions, we provide analytic continuation for the series $…
A congruence sum and rational approximations
Sandro Bettin
We give a reciprocity formula for a two-variable sum where the variables satisfy a linear congruence condition. We also prove that such sum is a measure of how well a rational is a…