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Gérald Tenenbaum

Publications and source records attributed to Gérald Tenenbaum.

At least 37 records · Page 2Linked to original sources
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.

math.NT↗
arXiv2022

The saddle-point method for general partition functions

We apply the saddle-point method to derive asymptotic estimates or asymptotic series for the number of partitions of a natural integer into parts chosen from a subset of the positive integers whose associated Dirichlet series satisfies certain analytic properties. This enables grouping in a single statement many cases studied in the literature, as well as a number of new ones.

math.CO↗
arXiv2022

Sur le biais d'une loi de probabilité relative aux entiers friables

The standard probability law on the set $S(x,y)$ of $y$-friable integers not exceeding $x$ assigns to each friable integer $n$ a probability proportional to $1/n^α$ where $α=α(x,y)$ is the saddle-point of the inverse Laplace integral for $Ψ(x,y):=|S(x,y)|$. This law presents a structural bias inasmuch it weights integers $>x$. We propose a quantitative measure of this bias and exhibit a related Gaussian distribution.

math.NT↗
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.

math.NT↗
arXiv2021

Multiplicative functions in large arithmetic progressions and applications

We establish new Bombieri-Vinogradov type estimates for a wide class of multiplicative arithmetic functions and derive several applications, including: a new proof of a recent estimate by Drappeau and Topacogullari for arithmetical correlations; a theorem of Erd{\H o}s-Wintner type with support equal to the level set of an additive function at shifted argument; and a law of iterated logarithm for the distribution of prime factors of integers weighted by $τ(n-1)$ where $τ$ denotes the divisor function.

math.NT↗
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.

math.PR↗
arXiv2020

Effective Erd\H os--Wintner theorems

The classical theorem of Erd\H os \& Wintner furnishes a criterion for the existence of a limiting distribution for a real, additive arithmetical function. This work is devoted to providing an effective estimate for the remainder term under the assumption that the conditions in the criterion are fulfilled. We also investigate the case of a conditional distribution.

math.NT↗
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.

math.NT↗
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 $(\log x)^{2+\varepsilon}\leqslant y\leqslant x$.

math.NT↗
arXiv2019

Power partitions and saddle-point method

For $k\geqslant 1$, denote by $p_k(n)$ the number of partitions of an integer $n$ into $k$-th powers. In this note, we apply the saddle-point method to provide a new proof for the well-known asymptotic expansion of $p_k(n)$. This approach turns out to significantly simplify those of Wright (1934), Vaughan (2015) and Gafni (2016).

math.NT↗
arXiv2019

Generalized Mertens sums

Extending a classical estimate of Mertens for the sum of the reciprocals of the first primes, we provide an explicit remainder formula for products of an arbitrary, but fixed, number of primes.

math.NT↗
arXiv2019

Remarques sur une somme liée à la fonction de Möbius

For integer $n\geqslant 1$ and real number $z\geqslant 1$, define $M(n,z):=\sum_{d|n,\,d\leqslant z}μ(d)$ where $μ$ denotes the Möbius function. Put ${\cal L}(y):=\exp\left\{(\log y)^{3/5}/(\log_2y)^{1/5}\right\}$ $(y\geqslant 3)$. We show that, for a suitable, explicit, constant $L>0$ and some absolute $c>0$, we have $S(x,z)= Lx+O\left({x/{\cal L}(3ξ)^c}\right)$ uniformly for $x\geqslant 1$, $ξ\leqslant z\leqslant x/ξ$.

math.NT↗
arXiv2019

A note on the normal largest gap between prime factors

Let $\{p_j(n)\}_{j=1}^{ω(n)}$ denote the increasing sequence of distinct prime factors of an integer $n$. We provide details for the proof of a statement of Erdős implying that, for any function $ξ(n)$ tending to infinity with $n$, we have $$f(n):=\max_{1\leqslant j<ω(n)}\log \Big({\log p_{j+1}(n)\over \log p_j(n)}\Big)=\log_3n+O(ξ(n))$$ for almost all integers $n$.

math.NT↗
arXiv2018

Somme des chiffres et changement de base

Let $s_a(n)$ denote the sum of digits of an integer $n$ in the base $a$ expansion. Answering, in a extended form, a question of Deshouillers, Habsieger, Laishram, and Landreau, we show that, provided $a$ and $b$ are multiplicatively independent, any positive real number is a limit point of the sequence $\{s_b(n)/s_a(n)\}_{n=1}^{\infty}$. We also provide upper and lower bounds for the counting functions of the corresponding subsequences.

math.NT↗
arXiv2018

Note sur les lois locales conjointes de la fonction nombre de facteurs premiers

Let $α\in]0,1]$ and let $Q_j$ $(1\leqslant j\leqslant r)$ denote distinct irreducible polynomials with integer coefficients. We show that, for vectors with coordinates not exceeding a constant multiple of their mean, the joint local distribution of the number of prime factors of the $Q_j(n)$ for $x<n\leqslant x+x^α$ is majorized by a constant multiple of the pairwise independency model, and we provide an upper bound for the constant in terms of the coefficients of the $Q_j$.

math.NT↗
arXiv2017

A remark on Sarnak's conjecture

We investigate Sarnak's conjecture on the Möbius function in the special case when the test function is the indicator of the set of integers for which a real additive function assumes a given value.

math.NT↗