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Gérald Tenenbaum

Publications and source records attributed to Gérald Tenenbaum.

At least 19 recordsLinked to original sources
arXiv2026

On partial derivatives of some summatory functions

Let $f$ be a real arithmetic function and let $g:[1,\infty[\to{\mathbb R}$ be a smooth function. We describe two emblematic instances in which saddle-point estimates may be used to evaluate the frequency, on the set of integers $n\leqslant x$, of the event $\{f(n)\leqslant g(n)\}$ from those relevant to the event $\{f(n)\leqslant y\}$. The first example revisits Dickman's historical contribution to the theory of friable integers. The second is concerned with the distribution of the squarefree kernel of an integer.

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arXiv2026

On a family of arithmetic series related to the Möbius function

Let $P^-(n)$ denote the smallest prime factor of a natural integer $n>1$. Furthermore let $μ$ and $ω$ denote respectively the Möbius function and the number of distinct prime factors function. We show that, given any set ${\scr P}$ of prime numbers with a natural density, we have $\sum_{P^-(n)\in \scr P}μ(n)ω(n)/n=0$ and provide a effective estimate for the rate of convergence. This extends a recent result of Alladi and Johnson, who considered the case when ${\scr P}$ is an arithmetic progression.

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arXiv2026

On integral boxes of minimal surface

Generalising the two-dimensional case, we provide estimates for the mean-values of the lengths of the edges of an integral box with given volume and minimal surface.

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arXiv2025

On effective mean-values of arithmetic functions

Let $r,\,f$ be multiplicative functions with $r\geqslant 0$, $f$ is complex valued, $|f|\leqslant r$, and $r$ satisfies some standard growth hypotheses. Let $x$ be large, and assume that, for some real number $τ$, the quantities $r(p)-\Re\{f(p)/p^{iτ}\}$ are small in various appropriate average senses over the set of prime numbers not exceeding $x$. We derive from recent effective mean-value estimates an effective comparison theorem between the mean-values of $f$ and of $r$ on the set of integers $\leqslant x$. We also provide effective estimates for certain weighted moments of additive functions and for sifted mean-values of non-negative multiplicative functions.

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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

Moyennes effectives de fonctions multiplicatives complexes

We establish effective mean-value estimates for a wide class of multiplicative arithmetic functions, thereby providing (essentially optimal) quantitative versions of Wirsing's classical estimates and extending those of Halász. Several applications are derived, including: estimates for the difference of mean-values of so-called pretentious functions, local laws for the distribution of prime factors in an arbitrary set, and weighted distribution of additive functions.

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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

Moyennes de certaines fonctions multiplicatives sur les entiers friables, 3

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 known results in the literature regarding both the quality of the error term and the range of validity. The one-sided hypotheses correspond to classical sieve assumptions. They are applied to provide an effective form of the Johnsen--Selberg prime power sieve.

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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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arXiv2024

On the friable mean-value of the Erdős-Hooley Delta function

For integer $n$ and real $u$, define $Δ(n,u):= |\{d : d \mid n,\,{\rm e}^u <d\leqslant {\rm e}^{u+1} \}|$. Then, put $ Δ(n):=\max_{u\in{\mathbb R}} Δ(n,u).$ We provide uniform upper and lower bounds for the mean-value of $Δ(n)$ over friable integers, i.e. integers free of large prime factors.

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arXiv2023

An Erdős-Kac theorem for integers with dense divisors

We show that for large integers $n$, whose ratios of consecutive divisors are bounded above by an arbitrary constant, the number of prime factors follows an approximate normal distribution, with mean $C \log_2 n$ and variance $V \log_2 n$, where $C=1/(1-e^{-γ})\approx 2.280$ and $V\approx 0.414$. This result is then generalized in two different directions.

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arXiv2023

On simply normal numbers with digit dependencies

Given an integer $b\geqslant 2$ and a set $P$ of prime numbers, the set $T_P $ of Toeplitz numbers comprises all elements of $[0,b[$ whose digits $(a_n)_{n\geqslant 1}$ in the base-$b$ expansion satisfy $a_n=a_{pn}$ for all $p\in P$ and $n\geqslant 1$. Using a completely additive arithmetical function, we construct a number in~$T_P$ that is simply Borel normal if, and only if, $\sum_{p\not \in P} 1/p=\infty$. We then provide an effective bound for the discrepancy.

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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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