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Cécile Dartyge

Publications and source records attributed to Cécile Dartyge.

9 recordsLinked to original sources
arXiv2026

On the digits of the sum of proper divisors

We study several probabilistic questions concerning the digits of $s(n)$, the sum of proper divisors of an integer $n$. In particular, we show that $s(n)$ obeys Benford's law with respect to logarithmic density. Moreover, we show that, for every function $k(x) \rightarrow \infty$, almost all integers $n \leq x$ have every decimal digit occurring among the first $k(x)$ digits and the last $k(x)$ digits of $s(n)$. We also present an upper bound for the number of composite integers $n$ up to $x$ for which $s(n)$ is missing at least one digit in its decimal expansion. This is in contrast with the main result of a recent paper of Benli, Cesana, Dartyge, Dombrowsky, and Thompson, in which the inputs $n$ were not required to be composite. It turns out that the primes make a substantial contribution to the preimage set $s^{-1}(\mathcal{A})$, where $\mathcal{A}$ is a set of integers with missing digits. Our result for composite $n$ shows that the count is much smaller when prime inputs are excluded.

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arXiv2025

Prime numbers with an almost prime reverse

Let $b$ be an integer greater than or equal to $2$. For any integer $n\in \left[b^{λ-1}, b^λ-1\right]$, we denote by $R_λ(n)$ the reverse of $n$ in base $b$, obtained by reversing the order of the digits of $n$. We establish a Bombieri-Vinogradov type theorem for the set of the reverses of the prime numbers. Combined with sieve methods, this permits us to prove that there exist $Ω_b\in\mathbb{N}$ and $c_b>0$ such that, for at least $c_b b^λ λ^{-2}$ primes $p\in \left[b^{λ-1}, b^λ-1\right]$, the reverse $R_λ(p)$ has at most $Ω_b$ prime factors. Some explicit admissible values of $Ω_b$ are given.

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arXiv2023

Reversible primes

For an $n$-bit positive integer $a$ written in binary as $$ a = \sum_{j=0}^{n-1} \varepsilon_{j}(a) \,2^j $$ where, $\varepsilon_j(a) \in \{0,1\}$, $j\in\{0, \ldots, n-1\}$, $\varepsilon_{n-1}(a)=1$, let us define $$ \overleftarrow{a} = \sum_{j=0}^{n-1} \varepsilon_j(a)\,2^{n-1-j}, $$ the digital reversal of $a$. Also let $\mathcal{B}_n = \{2^{n-1}\leq a<2^n:~a \text{ odd}\}.$ With a sieve argument, we obtain an upper bound of the expected order of magnitude for the number of $p \in \mathcal{B}_n$ such that $p$ and $\overleftarrow{p}$ are prime. We also prove that for sufficiently large $n$, $$ \left|\{a \in \mathcal{B}_n:~ \max \{Ω(a), Ω(\overleftarrow{a})\}\le 8 \}\right| \ge c\, \frac{2^n}{n^2}, $$ where $Ω(n)$ denotes the number of prime factors counted with multiplicity of $n$ and $c > 0$ is an absolute constant. Finally, we provide an asymptotic formula for the number of $n$-bit integers $a$ such that $a$ and $\overleftarrow{a}$ are both squarefree. Our method leads us to provide various estimates for the exponential sum $$ \sum_{a \in \mathcal{B}_n} \exp\left(2πi (αa + \vartheta \overleftarrow{a})\right) \quad(α,\vartheta \in\mathbb{R}). $$

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arXiv2023

Sums of proper divisors with missing digits

Let $s(n)$ denote the sum of proper divisors of an integer $n$. In 1992, Erdős, Granville, Pomerance, and Spiro (EGPS) conjectured that if $\mathcal{A}$ is a set of integers with asymptotic density zero then $s^{-1}(\mathcal{A})$ also has asymptotic density zero. In this paper we show that the EGPS conjecture holds when $\mathcal{A}$ is taken to be a set of integers with missing digits. In particular, we give a sharp upper bound for the size of this preimage set. We also provide an overview of progress towards the EGPS conjecture and survey recent work on sets of integers with missing digits.

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arXiv2021

On the distribution of the Rudin-Shapiro function for finite fields

Let $q=p^r$ be the power of a prime $p$ and $(β_1,\ldots ,β_r)$ be an ordered basis of $\mathbb{F}_q$ over $\mathbb{F}_p$. For $$ ξ=\sum\limits_{j=1}^r x_jβ_j\in \mathbb{F}_q \quad \mbox{with digits }x_j\in\mathbb{F}_p, $$ we define the Rudin-Shapiro function $R$ on $\mathbb{F}_q$ by $$ R(ξ)=\sum\limits_{i=1}^{r-1} x_ix_{i+1}, \quad ξ\in \mathbb{F}_q. $$ For a non-constant polynomial $f(X)\in \mathbb{F}_q[X]$ and $c\in \mathbb{F}_p$ we study the number of solutions $ξ\in \mathbb{F}_q$ of $R(f(ξ))=c$. If the degree $d$ of $f(X)$ is fixed, $r\ge 6$ and $p\rightarrow \infty$, the number of solutions is asymptotically $p^{r-1}$ for any $c$. The proof is based on the Hooley-Katz Theorem.

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arXiv2020

Entiers ultrafriables en progressions arithmétiques

A natural integer is called $y$-ultrafriable if none of the prime powers occurring in its canonical decomposition exceed $y$. We investigate the distribution of $y$-ultrafriable integers not exceeding $x$ among arithmetic progressions to the modulus $q$. Given a sufficiently small, positive constant $\varepsilon$, we obtain uniform estimates valid for $q\leqslant y^{c/\log_2y}$ whenever $\log y\leqslant (\log x)^\varepsilon$, and for $q\leqslant \sqrt{y}$ if $(\log x)^{2+\varepsilon}\leqslant y\leqslant x$.

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arXiv2019

Exponential sums with reducible polynomials

Hooley proved that if $f\in \Bbb Z [X]$ is irreducible of degree $\ge 2$, then the fractions $\{ r/n\}$, $0<r<n$ with $f(r)\equiv 0\pmod n$, are uniformly distributed in $(0,1)$. In this paper we study such problems for reducible polynomials of degree $2$ and $3$ and for finite products of linear factors. In particular, we establish asymptotic formulas for exponential sums over these normalized roots.

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arXiv2013

Sur la complexité de familles d'ensembles pseudo-aléatoires

In this paper we are interested in the following problem. Let $p$ be a prime number, $S\subset \F_p$ and $\cP\subset \{P\in\F_p [X]:°P\le d\}$. What is the largest integer $k$ such that for all subsets $\cA, \cB$ of $\F_p$ satisfying $\cA\cap\cB =\emptyset$ and $|\cA\cup\cB |=k$, there exists $P\in\cP$ such that $P(x)\in S$ if $x\in\cA$ and $P(x)\not\in S$ if $x\in\cB$? This problem corresponds to the study of the complexity of some families of pseudo-random subsets. First we recall this complexity definition and the context of pseudo-random subsets. Then we state the different results we have obtained according to the shape of the sets $S$ and $\cP$ considered. Some proofs are based on upper bounds for exponential sums or characters sums in finite fields, other proofs use combinatorics and additive number theory.

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