Abstract Localized large sums of random variables (2008)
We study large partial sums, localized with respect to the sums of variances, of a sequence of centered random variables. An application is given to the distribution of prime factors of typical...
The distribution of integers with at least two divisors in a short interval, Quart (2008)
Abstract. We estimate the density of integers which have more than one divisor in an interval (y, z] with z ≈ y + y/(log y) log 4−1. As a consequence, we determine the precise range of z such...
Moyennes de certaines fonctions multiplicatives sur les entiers friables, 2 (2008)
Hanrot, Guillaume, Tenenbaum, Gérald, Wu, Jie
Nous évaluons les moyennes indiquées dans le titre sous des hypothèses analytiques concernant la série de Dirichlet associée
Moyennes de certaines fonctions multiplicatives sur les entiers friables, 2 (2008)
Hanrot, Guillaume, Tenenbaum, Gérald, Wu, Jie
Nous évaluons les moyennes indiquées dans le titre sous des hypothèses analytiques concernant la série de Dirichlet associée
Moyennes de certaines fonctions multiplicatives sur les entiers friables, 3 (2008)
We consider logarithmic averages, over friable integers, of non-negative multiplicative functions. Under logarithmic, one-sided or two-sided hypotheses, we obtain sharp estimates that improve upon...
Moyennes de certaines fonctions multiplicatives sur les entiers friables, 3 (2008)
We consider logarithmic averages, over friable integers, of non-negative multiplicative functions. Under logarithmic, one-sided or two-sided hypotheses, we obtain sharp estimates that improve upon...
Moyennes de certaines fonctions multiplicatives sur les entiers friables, 2 (2008)
Hanrot, Guillaume, Tenenbaum, Gérald, Wu, Jie
We derive new, very precise estimates for averages of arithmetic functions over friable integers from analytic information on their associated Dirichlet series. These yield significant improvements...
Entiers friables: inégalité de Turán-Kubilius et applications (2007)
We generalize the classical Turán-Kubilus inequality to friable integers and obtain sharp estimates for the involved constant in large subdomains. Several applications are derived in probabilitic...
Moyennes de certaines fonctions multiplicatives sur les entiers friables, 2 (2007)
Hanrot, Guillaume, Tenenbaum, Gérald, Wu, Jie
Nous évaluons les moyennes indiquées dans le titre sous des hypothèses analytiques concernant la série de Dirichlet associée
THE DISTRIBUTION OF INTEGERS WITH AT LEAST TWO DIVISORS IN A SHORT INTERVAL (2007)
Ford, Kevin, Tenenbaum, Gérald
We estimate the density of integers which have more than one divisor in an interval (y, z] with z ≈ y + y/(log y)log 4 − 1. As a consequence, we determine the precise range of z such that most...
Moyennes de certaines fonctions multiplicatives sur les entiers friables, 3 (2007)
We consider logarithmic averages, over friable integers, of non-negative multiplicative functions. Under logarithmic, one-sided or two-sided hypotheses, we obtain sharp estimates that improve upon...
Remarques sur les valeurs moyennes de fonctions multiplicatives (2007)
By a self-contained approach, we show that a real valued multiplicative function whose square has a mean-value must itself have a mean value, and we provide a sufficient condition for the vanishing...
Moyennes de certaines fonctions multiplicatives sur les entiers friables, 3 (2007)
We consider logarithmic averages, over friable integers, of non-negative multiplicative functions. Under logarithmic, one-sided or two-sided hypotheses, we obtain sharp estimates that improve upon...
Moyennes de certaines fonctions multiplicatives sur les entiers friables, 2 (2007)
Hanrot, Guillaume, Tenenbaum, Gérald, Wu, Jie
Nous évaluons les moyennes indiquées dans le titre sous des hypothèses analytiques concernant la série de Dirichlet associée
Entiers friables: inégalité de Turán-Kubilius et applications (2007)
We generalize the classical Turán-Kubilus inequality to friable integers and obtain sharp estimates for the involved constant in large subdomains. Several applications are derived in probabilitic...
Moyennes de certaines fonctions multiplicatives sur les entiers friables, 2 (2007)
Hanrot, Guillaume, Tenenbaum, Gérald, Wu, Jie
Nous évaluons les moyennes indiquées dans le titre sous des hypothèses analytiques concernant la série de Dirichlet associée
Moyennes de certaines fonctions multiplicatives sur les entiers friables, 3 (2007)
We consider logarithmic averages, over friable integers, of non-negative multiplicative functions. Under logarithmic, one-sided or two-sided hypotheses, we obtain sharp estimates that improve upon...
Remarques sur les valeurs moyennes de fonctions multiplicatives (2007)
By a self-contained approach, we show that a real valued multiplicative function whose square has a mean-value must itself have a mean value, and we provide a sufficient condition for the vanishing...
Moyennes de certaines fonctions multiplicatives sur les entiers friables, 3 (2007)
We consider logarithmic averages, over friable integers, of non-negative multiplicative functions. Under logarithmic, one-sided or two-sided hypotheses, we obtain sharp estimates that improve upon...
An overlapping theorem with applications (2007)
Cilleruelo, Javier, Tenenbaum, Gérald
We establish a general and optimal lower bound for the complete sum of the probabilities of k-intersections of n events. We then describe various applications to additive and multiplicative number...
Moyennes de certaines fonctions multiplicatives sur les entiers friables, 4 (2007)
We provide two generic applications of previously established estimates for friable mean values of non-negative multiplicative functions.
Remarques sur les valeurs moyennes de fonctions multiplicatives (2007)
By a self-contained approach, we show that a real valued multiplicative function whose square has a mean-value must itself have a mean value, and we provide a sufficient condition for the vanishing...
Remarques sur les valeurs moyennes de fonctions multiplicatives (2007)
By a self-contained approach, we show that a real valued multiplicative function whose square has a mean-value must itself have a mean value, and we provide a sufficient condition for the vanishing...
Moyennes de certaines fonctions multiplicatives sur les entiers friables, 4 (2007)
We provide two generic applications of previously established estimates for friable mean values of non-negative multiplicative functions.
An overlapping theorem with applications (2007)
Cilleruelo, Javier, Tenenbaum, Gérald
We establish a general and optimal lower bound for the complete sum of the probabilities of k-intersections of n events. We then describe various applications to additive and multiplicative number...
Localized large sums of random variables (2006)
Ford, Kevin, Tenenbaum, Gérald
We study large partial sums, localized with respect to the sums of variances, of a sequence of centered random variables. An application is given to the distribution of prime factors of typical...
Moyennes de certaines fonctions multiplicatives sur les entiers friables, 2 (2006)
Hanrot, Guillaume, Tenenbaum, Gérald, Wu, Jie
Nous évaluons les moyennes indiquées dans le titre sous des hypothèses analytiques concernant la série de Dirichlet associée
Sur le comportement local de la répartition de lindicatrice dEuler (2006)
Tenenbaum, Gérald, Toulmonde, Vincent
Let phi denote Euler's totient function. A classical result of Schoenberg asserts that G(t):=dens{n>= 1:varphi(n)/n
Integers with a large friable component (2006)
We provide sharp estimates for the number of integers not exceeding whose largest y-friable divisor exceeds z.
An overlapping theorem with applications (2006)
Cilleruelo, Javier, Tenenbaum, Gérald
Let $\mu$ be a positive measure on a set $\Omega$, $\{ E_j\}_{j=1}^k$ a family of measurable subsets, and set $$\tau_m:=\sum_{1\le j_1< \cdots
Localized large sums of random variables (2006)
Ford, Kevin, Tenenbaum, Gérald
We study large partial sums, localized with respect to the sums of variances, of a sequence of centered random variables. An application is given to the distribution of prime factors of typical...
The distribution of integers with at least two divisors in a short interval (2006)
Ford, Kevin, Tenenbaum, Gérald
We determine the true order of magnitude of the number of integers not exceeding x which have a prescribed number of divisors in a short interval, precisely defined in the text.
Localized large sums of random variables (2006)
Ford, Kevin, Tenenbaum, Gérald
We study large partial sums, localized with respect to the sums of variances, of a sequence of centered random variables. An application is given to the distribution of prime factors of typical...
The distribution of integers with at least two divisors in a short interval (2006)
Ford, Kevin, Tenenbaum, Gérald
We determine the true order of magnitude of the number of integers not exceeding x which have a prescribed number of divisors in a short interval, precisely defined in the text.
Integers with a large friable component (2006)
We provide sharp estimates for the number of integers not exceeding whose largest y-friable divisor exceeds z.
An overlapping theorem with applications (2006)
Cilleruelo, Javier, Tenenbaum, Gérald
Let $\mu$ be a positive measure on a set $\Omega$, $\{ E_j\}_{j=1}^k$ a family of measurable subsets, and set $$\tau_m:=\sum_{1\le j_1< \cdots
Moyennes de certaines fonctions multiplicatives sur les entiers friables, 2 (2006)
Hanrot, Guillaume, Tenenbaum, Gérald, Wu, Jie
Nous évaluons les moyennes indiquées dans le titre sous des hypothèses analytiques concernant la série de Dirichlet associée
Integers with a large friable component (2006)
We provide sharp estimates for the number of integers not exceeding whose largest y-friable divisor exceeds z.
An overlapping theorem with applications (2006)
Cilleruelo, Javier, Tenenbaum, Gérald
Let $\mu$ be a positive measure on a set $\Omega$, $\{ E_j\}_{j=1}^k$ a family of measurable subsets, and set $$\tau_m:=\sum_{1\le j_1< \cdots
Propriétés statistiques des entiers friables (2006)
De La Bretèche, Régis, Tenenbaum, Gérald
Let Psi(x,y) (resp. Psi_m(x,y)) denote the number of integers not exceeding x that are y-friable, i.e., have no prime factor exceeding y (resp. and are coprime to m). Evaluating the ratio...
Congruences de sommes de chiffres de valeurs polynomiales (2006)
Dartyge, Cécile, Tenenbaum, Gérald
Let $s_q(n)$ denote the sum of digits of $n$ in the $q$-ary digital expansion. Given a polynomial $f$ with integer coefficients, degree $d\ge 1$, it is shown that there exists $C>0$ such thatfor any...
Congruences de sommes de chiffres de valeurs polynomiales (2006)
Dartyge, Cécile, Tenenbaum, Gérald
Let m, g, q be integers with q >= 2 and (m,q-1)=1. For natural integer n, denote by s_q(n) the sum of digits of n in the q-ary digital expansion. Given a polynomial f with integer coefficients,...
Congruences de sommes de chiffres de valeurs polynomiales (2006)
Dartyge, Cécile, Tenenbaum, Gérald
Let m, g, q be integers with q >= 2 and (m,q-1)=1. For natural integer n, denote by s_q(n) the sum of digits of n in the q-ary digital expansion. Given a polynomial f with integer coefficients,...
Sur le comportement local de la répartition de l'indicatrice d'Euler (2006)
Tenenbaum, Gérald, Toulmonde, Vincent
Let phi denote Euler's totient function. A classical result of Schoenberg asserts that G(t):=dens{n>= 1:varphi(n)/n
The distribution of integers with at least two divisors in a short interval (2006)
Ford, Kevin, Tenenbaum, Gérald
We determine the true order of magnitude of the number of integers not exceeding x which have a prescribed number of divisors in a short interval, precisely defined in the text.
Congruences de sommes de chiffres de valeurs polynomiales (2006)
Dartyge, Cécile, Tenenbaum, Gérald
Let m, g, q be integers with q >= 2 and (m,q-1)=1. For natural integer n, denote by s_q(n) the sum of digits of n in the q-ary digital expansion. Given a polynomial f with integer coefficients,...
Sur le comportement local de la répartition de l'indicatrice d'Euler (2006)
Tenenbaum, Gérald, Toulmonde, Vincent
Let phi denote Euler's totient function. A classical result of Schoenberg asserts that G(t):=dens{n>= 1:varphi(n)/n
Integers with a large friable component (2006)
We provide sharp estimates for the number of integers not exceeding whose largest y-friable divisor exceeds z.
Localized large sums of random variables (2006)
Ford, Kevin, Tenenbaum, Gérald
We study large partial sums, localized with respect to the sums of variances, of a sequence of centered random variables. An application is given to the distribution of prime factors of typical...
The distribution of integers with at least two divisors in a short interval (2006)
Ford, Kevin, Tenenbaum, Gérald
We determine the true order of magnitude of the number of integers not exceeding x which have a prescribed number of divisors in a short interval, precisely defined in the text.
Integers with a large friable component (2006)
We provide sharp estimates for the number of integers not exceeding whose largest y-friable divisor exceeds z.
Localized large sums of random variables (2006)
Ford, Kevin, Tenenbaum, Gérald
We study large partial sums, localized with respect to the sums of variances, of a sequence of centered random variables. An application is given to the distribution of prime factors of typical...
Sur le comportement local de la répartition de l'indicatrice d'Euler (2006)
Tenenbaum, Gérald, Toulmonde, Vincent
Let phi denote Euler's totient function. A classical result of Schoenberg asserts that G(t):=dens{n>= 1:varphi(n)/n
Congruences de sommes de chiffres de valeurs polynomiales (2006)
Dartyge, Cécile, Tenenbaum, Gérald
Let m, g, q be integers with q >= 2 and (m,q-1)=1. For natural integer n, denote by s_q(n) the sum of digits of n in the q-ary digital expansion. Given a polynomial f with integer coefficients,...
Entiers friables: inégalité de Turán-Kubilius et applications (2005)
We generalize the classical Turán-Kubilus inequality to friable integers and obtain sharp estimates for the involved constant in large subdomains. Several applications are derived in probabilitic...
Sur l'écart quadratique moyen des diviseurs d'un entier normal, 2 (2005)
As in the first part of this work with A. Raouj [Math. Proc. Cambridge Philos. Soc. 126 (1999), no. 3, 399--415], this work is devoted to a measure of the propensity of an integer's divisors to...
On products of ratios of consecutive integers (2005)
De La Bretèche, Régis, Pomerance, Carl, Tenenbaum, Gérald
Take the product of the numbers (n/(n+1))^{\epsilon_n} for 1
Inégalités d'Erdös-Turán (2005)
Rivat, Joël, Tenenbaum, Gérald
The Erdös-Turán inequality measures the distance from uniform distribution of any given sequence on the torus as a function of an arbitrary parameter and two constants, $c_1$ and $c_2$. We show...
Ramdani, Karim, Takahashi, Takeo, Tenenbaum, Gérald, Tucsnak, Marius
Let $A$ be a possibly unbounded skew-adjoint operator on the Hilbert space $X$ with compact resolvent. Let $C$ be a bounded operator from $\Dscr(A)$ to another Hilbert space $Y$. We consider the...
Sommes des chiffres de multiples d'entiers (2005)
Dartyge, Cécile, Tenenbaum, Gérald
Let $s_q(n)$ denote the sum of digits of $n$ in the $q$-ary digital expansion. We give upper bounds for exponential sums like $$G(x,y,\theta ,\alpha , {\bf h})=\sum_{x
Sommes des chiffres de multiples d'entiers (2005)
Dartyge, Cécile, Tenenbaum, Gérald
Let q in N, q >= 2. For n in N, denote by s_q(n) the sum of digits of n in the q-ary digital expansion. We give upper bounds for exponential sums like G(x,y, theta ; a,h) = sum_{x= 2 by Coquet and...
Sommes des chiffres de multiples d'entiers (2005)
Dartyge, Cécile, Tenenbaum, Gérald
Let q in N, q >= 2. For n in N, denote by s_q(n) the sum of digits of n in the q-ary digital expansion. We give upper bounds for exponential sums like G(x,y, theta ; a,h) = sum_{x= 2 by Coquet and...
Ramdani, Karim, Takahashi, Takeo, Tenenbaum, Gérald, Tucsnak, Marius
Let $A$ be a possibly unbounded skew-adjoint operator on the Hilbert space $X$ with compact resolvent. Let $C$ be a bounded operator from $\Dscr(A)$ to another Hilbert space $Y$. We consider the...
Sur l'écart quadratique moyen des diviseurs d'un entier normal, 2 (2005)
As in the first part of this work with A. Raouj [Math. Proc. Cambridge Philos. Soc. 126 (1999), no. 3, 399--415], this work is devoted to a measure of the propensity of an integer's divisors to...
On products of ratios of consecutive integers (2005)
De La Bretèche, Régis, Pomerance, Carl, Tenenbaum, Gérald
Take the product of the numbers (n/(n+1))^{\epsilon_n} for 1
Inégalités d'Erdös-Turán (2005)
Rivat, Joël, Tenenbaum, Gérald
The Erdös-Turán inequality measures the distance from uniform distribution of any given sequence on the torus as a function of an arbitrary parameter and two constants, $c_1$ and $c_2$. We show...
Propriétés statistiques des entiers friables (2005)
De La Bretèche, Régis, Tenenbaum, Gérald
Let Psi(x,y) (resp. Psi_m(x,y)) denote the number of integers not exceeding x that are y-friable, i.e., have no prime factor exceeding y (resp. and are coprime to m). Evaluating the ratio...
Entiers friables: inégalité de Turán-Kubilius et applications (2005)
We generalize the classical Turán-Kubilus inequality to friable integers and obtain sharp estimates for the involved constant in large subdomains. Several applications are derived in probabilitic...
Sommes des chiffres de multiples d'entiers (2005)
Dartyge, Cécile, Tenenbaum, Gérald
Let q in N, q >= 2. For n in N, denote by s_q(n) the sum of digits of n in the q-ary digital expansion. We give upper bounds for exponential sums like G(x,y, theta ; a,h) = sum_{x= 2 by Coquet and...
Ramdani, Karim, Takahashi, Takeo, Tenenbaum, Gérald, Tucsnak, Marius
Let $A$ be a possibly unbounded skew-adjoint operator on the Hilbert space $X$ with compact resolvent. Let $C$ be a bounded operator from $\Dscr(A)$ to another Hilbert space $Y$. We consider the...
Sur l'écart quadratique moyen des diviseurs d'un entier normal, 2 (2005)
As in the first part of this work with A. Raouj [Math. Proc. Cambridge Philos. Soc. 126 (1999), no. 3, 399--415], this work is devoted to a measure of the propensity of an integer's divisors to...
On products of ratios of consecutive integers (2005)
De La Bretèche, Régis, Pomerance, Carl, Tenenbaum, Gérald
Take the product of the numbers (n/(n+1))^{\epsilon_n} for 1
Constantes d'Erdös-Turán (2005)
Rivat, Joël, Tenenbaum, Gérald
The Erdös-Turán inequality measures the distance from uniform distribution of any given sequence on the torus as a function of an arbitrary parameter and two constants, $c_1$ and $c_2$. We show...
Propriétés statistiques des entiers friables (2005)
De La Bretèche, Régis, Tenenbaum, Gérald
Let Psi(x,y) (resp. Psi_m(x,y)) denote the number of integers not exceeding x that are y-friable, i.e., have no prime factor exceeding y (resp. and are coprime to m). Evaluating the ratio...
Entiers friables: inégalité de Turán-Kubilius et applications (2005)
We generalize the classical Turán-Kubilus inequality to friable integers and obtain sharp estimates for the involved constant in large subdomains. Several applications are derived in probabilitic...
Entiers friables: inégalité de Turán-Kubilius et applications (2005)
We generalize the classical Turán-Kubilus inequality to friable integers and obtain sharp estimates for the involved constant in large subdomains. Several applications are derived in probabilitic...
Constantes d'Erdös-Turán (2005)
Rivat, Joël, Tenenbaum, Gérald
The Erdös-Turán inequality measures the distance from uniform distribution of any given sequence on the torus as a function of an arbitrary parameter and two constants, $c_1$ and $c_2$. We show...
Propriétés statistiques des entiers friables (2005)
De La Bretèche, Régis, Tenenbaum, Gérald
Let Psi(x,y) (resp. Psi_m(x,y)) denote the number of integers not exceeding x that are y-friable, i.e., have no prime factor exceeding y (resp. and are coprime to m). Evaluating the ratio...
On products of ratios of consecutive integers (2005)
De La Bretèche, Régis, Pomerance, Carl, Tenenbaum, Gérald
Take the product of the numbers (n/(n+1))^{\epsilon_n} for 1
Sur l'écart quadratique moyen des diviseurs d'un entier normal, 2 (2005)
As in the first part of this work with A. Raouj [Math. Proc. Cambridge Philos. Soc. 126 (1999), no. 3, 399--415], this work is devoted to a measure of the propensity of an integer's divisors to...
Sommes des chiffres de multiples d'entiers (2005)
Dartyge, Cécile, Tenenbaum, Gérald
Let q in N, q >= 2. For n in N, denote by s_q(n) the sum of digits of n in the q-ary digital expansion. We give upper bounds for exponential sums like G(x,y, theta ; a,h) = sum_{x= 2 by Coquet and...
Ramdani, Karim, Takahashi, Takeo, Tenenbaum, Gérald, Tucsnak, Marius
Let $A$ be a possibly unbounded skew-adjoint operator on the Hilbert space $X$ with compact resolvent. Let $C$ be a bounded operator from $\Dscr(A)$ to another Hilbert space $Y$. We consider the...
Sur la répartition divisorielle normale de $\vartheta d\pmod 1$. (2004)
Kerner, Sébastien, Tenenbaum, Gérald
Let $\scr E$ denote the set of irrational numbers whose continued fraction convergents $p_j/q_j$ obey the rule $$\log q_{j+1} \le (\log q_j)^{1 + o(1)} \quad (j \to \infty).$$ Thus $\scr E$ contains...
Séries trigonométriques à coefficients arithmétiques (2004)
De La Bretèche, Régis, Tenenbaum, Gérald
By exploiting the recent analytic tool of friable summation, this work describes a new approach to a class of problems in multiplicative number theory and Fourier series theory originated by H....
Séries trigonométriques à coefficients arithmétiques (2004)
De La Bretèche, Régis, Tenenbaum, Gérald
By exploiting the recent analytic tool of friable summation, this work describes a new approach to a class of problems in multiplicative number theory and Fourier series theory originated by H....
Sur la répartition divisorielle normale de $\vartheta d\pmod 1$. (2004)
Kerner, Sébastien, Tenenbaum, Gérald
Let $\scr E$ denote the set of irrational numbers whose continued fraction convergents $p_j/q_j$ obey the rule $$\log q_{j+1} \le (\log q_j)^{1 + o(1)} \quad (j \to \infty).$$ Thus $\scr E$ contains...
Moyennes de certaines fonctions multiplicatives sur les entiers friables (2003)
An integer n is called friable when its prime number factorization consists exclusively of (relatively) small factors. Let P(n) be the largest prime factor occurring in this factorization, and let...
Moyennes de certaines fonctions multiplicatives sur les entiers friables (2003)
An integer n is called friable when its prime number factorization consists exclusively of (relatively) small factors. Let P(n) be the largest prime factor occurring in this factorization, and let...
Inversion de Laplace effective (2001)
Stef, André, Tenenbaum, Gérald
Let $F,G$ be arbitrary distribution functions on the real line and let $\widehat{F},\widehat{G}$ denote their respective bilateral Laplace transforms. Let $\kappa > 0$ and let $h : \mathbb{R}^+ \to...
We provide a simple proof of a result which generalises and makes slightly more precise the main theorem in the paper quoted in the title. This concerns weak convergence, in the Skorokhod space, of...
We provide a simple proof of a result which generalises and makes slightly more precise the main theorem in the paper quoted in the title. This concerns weak convergence, in the Skorokhod space, of...
We provide a simple proof of a result which generalises and makes slightly more precise the main theorem in the paper quoted in the title. This concerns weak convergence, in the Skorokhod space, of...
We provide a simple proof of a result which generalises and makes slightly more precise the main theorem in the paper quoted in the title. This concerns weak convergence, in the Skorokhod space, of...
Sur une question d'Erdös et Schinzel (1990)
Let F(n) denote a polynomial with integer coefficients and define H_F(x,y,z) to be the number of integers not exceeding x for which F(n) has at least one divisor d such that yxexp(c loglog x...
Sur une question d'Erdös et Schinzel (1990)
Let F(n) denote a polynomial with integer coefficients and define H_F(x,y,z) to be the number of integers not exceeding x for which F(n) has at least one divisor d such that y
Sur une question d'Erdös et Schinzel (1990)
Let F(n) denote a polynomial with integer coefficients and define H_F(x,y,z) to be the number of integers not exceeding x for which F(n) has at least one divisor d such that y
Sur une question d'Erdös et Schinzel (1990)
Let F(n) denote a polynomial with integer coefficients and define H_F(x,y,z) to be the number of integers not exceeding x for which F(n) has at least one divisor d such that y
Localized large sums of random variables
Ford, Kevin, Tenenbaum, Gérald
We study large partial sums, localized with respect to the sums of variances, of a sequence of centered random variables. An application is given to the distribution of prime factors of typical...