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Régis de la Bretèche

Publications and source records attributed to Régis de la Bretèche.

At least 19 recordsLinked to original sources
arXiv2026

Random Multiplicative Functions and Making Squares from Polynomial Values

For a large family of polynomials $P(X)\in \mathbb{Z}[X]$, we prove central limit theorems for $\sum_{n\le N} f(P(n))$ for both Rademacher and extended Rademacher multiplicative functions $f$. To achieve this, we establish a paucity phenomenon in counting solutions to \[P(n_1)P(n_2)P(n_3)P(n_4) = \square, \quad 1\le n_1, n_2, n_3, n_4 \le N.\] Results of Hooley, Evertse--Silverman, and Reuss play an important role in the proof. Our estimates are sharpest for $°P = 2$, thanks to the rich theory of Pell--Fermat equations.

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arXiv2025

Mean values of arithmetic functions and application to sums of powers

We provide new upper bounds for sums of certain arithmetic functions in many variables at polynomial arguments and, exploiting recent progress on the mean-value of the Erd\H os-Hooley $Δ$-function, we derive lower bounds for the cardinality of those integers not exceeding a given limit that are expressible as some sums of powers.

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arXiv2025

Remarks on the Selberg--Delange method

Let $\varrho$ be a complex number and let $f$ be a multiplicative arithmetic function whose Dirichlet series takes the form $ζ(s)^\varrho G(s)$, where $G$ is associated to a multiplicative function $g$. The classical Selberg-Delange method furnishes asymptotic estimates for averages of $f$ under assumptions of either analytic continuation for $G$, or absolute convergence of a finite number of derivatives of $G(s)$ at $s=1$. We consider different set of hypotheses, not directly comparable to the previous ones, and investigate how they can yield sharp asymptotic estimates for the averages of~$f$.

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arXiv2025

Note on a conjecture of Hildebrand regarding friable integers

Hildebrand proved that the smooth approximation for the number $Ψ(x,y)$ of $y$-friable integers not exceeding $x$ holds for $y>(\log x)^{2+\varepsilon}$ under the Riemann hypothesis and conjectured that it fails when $y\leqslant (\log x)^{2-\varepsilon}$. This conjecture has been recently confirmed by Gorodetsky by an intricate argument. We propose a short, straight-forward proof.

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arXiv2025

Note on the mean value of the Erdős--Hooley Delta-function

For integer $n\geqslant 1$ and real $u$, let $Δ(n,u):=|\{d:d\mid n,\,{\rm e}^u<d\leqslant {\rm e}^{u+1}\}|$. The Erdős--Hooley Delta-function is then defined by $Δ(n):=\max_{u\in{\mathbb R}}Δ(n,u).$ We improve a recent upper bound for the mean value of this function by showing that, for large $x$, we have $$\sum_{n\leqslant x}Δ(n)\ll x(\log_2x)^{ 5/2}.$$

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arXiv2024

Répartition conjointe de trois nombres premiers et applications

We prove asymptotic results for the singular series associated to the distribution of three primes. Assuming a quantitative version of Hardy and Littlewood's conjecture on prime 3-tuples, we deduce an asymptotic formula related to the joint distribution of three primes. This improves recent results of Kuperberg and completes results by Montgomery and Soundararajan. Following the Montgomery and Soundararajan approach, we derive conjectural applications to the distribution of primes in short intervals.

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arXiv2024

Friable averages of oscillating multiplicative functions

We evaluate friable averages of arithmetic functions whose Dirichlet series is analytically close to some negative power of the Riemann zeta function. We obtain asymptotic expansions resembling those provided by the Selberg-Delange method in the non-friable case. An application is given to summing truncated versions of such functions.

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arXiv2024

Friable averages of complex arithmetic functions

We evaluate friable averages of arithmetic functions whose Dirichlet series is analytically close to some complex power of the Riemann zeta function. We obtain asymptotic expansions resembling those provided by the Selberg-Delange method in the non-friable case. Some application are provided to the friable distribution of the additive function counting the total number of prime factors.

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arXiv2023

Moments of moments of primes in arithmetic progressions

We establish unconditional $Ω$-results for all weighted even moments of primes in arithmetic progressions. We also study the moments of these moments and establish lower bounds under GRH. Finally, under GRH and LI we prove an asymptotic for all moments of the associated limiting distribution, which in turn indicates that our unconditional and GRH results are essentially best possible. Using our probabilistic results, we formulate a conjecture on the moments with a precise associated range of validity, which we believe is also best possible. This last conjecture implies a $q$-analogue of the Montgomery-Soundararajan conjecture on the Gaussian distribution of primes in short intervals. The ideas in our proofs include a novel application of positivity in the explicit formula and the combinatorics of arrays of characters which are fixed by certain involutions.

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arXiv2023

Lower bounds on weighted moments of primes in short intervals in number fields

We consider an analog of a conjecture of Montgomery and Soundararajan on the moments of primes in short intervals in number fields; this analog was discussed and heuristically derived in a paper of the second author, Rodgers, and Roditty-Gershon. Adapting work of the first author and Fiorilli in the integer case, we establish lower bounds on a weighted version of these moments which agree with the conjectured values.

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arXiv2022

Two upper bounds for the Erdős--Hooley Delta-function

For integer $n\geqslant 1$ and real $u$, let $Δ(n,u):=|\{d:d\mid n,\,{\rm e}^u<d\leqslant {\rm e}^{u+1}\}|$. The Erdős--Hooley Delta-function is then defined by $Δ(n):=\max_{u\in{\mathbb R}}Δ(n,u).$ We improve the current upper bounds for the average and normal orders of this arithmetic function.

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arXiv2022

Sommes de Gál et applications

We evaluate the asymptotic size of various sums of Gál type, in particular $$S( \mathcal{M}):=\sum_{m,n\in\mathcal{M}} \sqrt{(m,n) \over [m,n]},$$ where $\mathcal{M}$ is a finite set of integers. Elaborating on methods recently developed by Bondarenko and Seip, we obtain an asymptotic formula for $$\log\Big( \sup_{|\mathcal{M}|= N}{S( \mathcal{M})/N}\Big)$$ and derive new lower bounds for localized extreme values of the Riemann zeta-function, for extremal values of some Dirichlet $L$-functions at $s=1/2$, and for large character sums.

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arXiv2021

On the gap distribution of prime factors

Let $\{p_j(n)\}_{j=1}^{ω(n)}$ denote the increasing sequence of distinct prime factors of an integer $n$. For $z\geqslant 0$, let $G(n;z)$ denote the number of those indexes $j$ such that $p_{j+1}(n)>p_j(n)^{\exp z}$. We show uniform convergence, with almost optimal effective estimate of the speed, of the distribution of $G(n;z)$ on $\{n:1\leqslant n\leqslant N\}$ to a Gaussian limit law with mean ${\rm e}^{-z}\log_2n$ and variance $\{{\rm e}^{-z}-2z{\rm e}^{-2z}\}\log_2n$, and we establish an asymptotic formula with remainder for all centered moments.

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arXiv2021

On strong and almost sure local limit theorems for a probabilistic model of the Dickman distribution

Let $\{Z_k\}_{k\geqslant 1}$ denote a sequence of independent Bernoulli random variables defined by ${\mathbb P}(Z_k=1)=1/k=1-{\mathbb P}(Z_k=0)$ $(k\geqslant 1)$ and put $T_n:=\sum_{1\leqslant k\leqslant n}kZ_k$. It is then known that $T_n/n$ converges weakly to a real random variable $D$ with density proportional to the Dickman function, defined by the delay-differential equation $u\varrho'(u)+\varrho(u-1)=0$ $(u>1)$ with initial condition $\varrho(u)=1$ $(0\leqslant u\leqslant 1)$. Improving on earlier work, we propose asymptotic formulae with remainders for the corresponding local and almost sure limit theorems.

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arXiv2020

On a conjecture of Montgomery and Soundararajan

We establish lower bounds for all weighted even moments of primes up to $X$ in intervals which are in agreement with a conjecture of Montgomery and Soundararajan. Our bounds hold unconditionally for an unbounded set of values of $X$, and hold for all $X$ under the Riemann Hypothesis. We also deduce new unconditional $Ω$-results for the classical prime counting function.

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arXiv2020

Small Gál sums and applications

In recent years, maximizing Gál sums regained interest due to a firm link with large values of $L$-functions. In the present paper, we initiate an investigation of small sums of Gál type, with respect to the $L^1$-norm. We also consider the intertwined question of minimizing weighted versions of the usual multiplicative energy. We apply our estimates to: (i) a logarithmic refinement of Burgess' bound on character sums, improving previous results of Kerr, Shparlinski and Yau; (ii) an improvement on earlier lower bounds by Louboutin and the second author for the number of non vanishing theta functions associated to Dirichlet characters; and (iii) new lower bounds for low moments of character sums.

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arXiv2019

On the number of products which form perfect powers and discriminants of multiquadratic extensions

We study some counting questions concerning products of positive integers $u_1, \ldots, u_n$ which form a non-zero perfect square, or more generally, a perfect $k$-th power. We obtain an asymptotic formula for the number of such integers of bounded size and in particular improve and generalize a result of D. I. Tolev (2011). We also use similar ideas to count the discriminants of number fields which are multiquadratic extensions of $\mathbb{Q}$ and improve and generalize a result of N. Rome (2017).

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