Searcharxiv⌕ Search

arXiv subjects

Eric Saias

Publications and source records attributed to Eric Saias.

10 recordsLinked to original sources
arXiv2025

Etude du graphe divisoriel 6

The divisor graph is the non oriented graph whose vertices are the positive integers, and edges are the {a,b} such that a divides b or b divides a. Let F(x,y) be the maximum number of integers<= x belonging in one of y pairwise disjoint simple path of the restriction of the divisor graph to integers <= x. Our main result is the following. There exist two real numbers K >c>0 such that for every x and y with x>=2y>=2 , we have cx / log(x/y) <= F(x,y) <= Kx / log(x/y). It answers a question of Erd\"os.

math.CO↗
arXiv2021

Etude du graphe divisoriel 5

The divisor graph is the non oriented graph whose vertices are the positive integers, and edges are the {a,b} such that a divides b. Let P(n) be the largest prime factor of n, S(x,y) = {n<=x: P(n) <= y} and Psi(x,y) = Card S(x,y). Let f(x,y) be the biggest number of vertices in a simple path of the divisor graph restricted to S(x,y). We give a lower bound of f(x,y) uniformly in 2 <= y <= x which improves a result of Tenenbaum. It gives f(x) := f(x,x) >= (c + o(1)) x/logx with c = 0.3 instead of c = 0.017 in the previous result. For the little y, it implies the new formula f(x,y) = (1+o(1)) Psi(x,y), when x goes to infinity and y = o(logx). Finally we prove also a lower bound for a variant of f(x,y), which will be used in a following paper to answer a question of Erdös.

math.NT↗
arXiv2021

Un exemple de somme de s\'erie de vecteurs propres \`a valeurs propres de module un, non r\'ecurrente

Let $\zeta^*(s)=\sum_{n=1}^{+\infty}(-1)^n/n^s$ and $\tau$ the operator defined on the Frechet space of holomorphic functions in $\{s\in \mathbb C :1/2< Re \, s<1\}$ by $\tau f(s)= f(s-2i\pi/\log 2)$. We show that the Riemann Hypothesis is equivalent to the strong recurrence of $\zeta^*(s)$ for $\tau$. It follows that a sufficient condition for $RH$ would be that every sum of a series of eigenvectors with unimodular eigenvalues for an operator $u$ is strongly recurrent for $u$. But we give a counterexample showing that it is not the case.

math.CV↗
arXiv2021

Sur le spectre des opérateurs rigides

A bounded operator $u$ on $X$ is called rigid when there is an increasing sequence of positive integers $(n_k)_{k\geq 1}$, such that for every $x$ in $X$ we have $\lim_{k \rightarrow +\infty} u^{n_k} x = x$. For any $r$ in $[0,1]$, we construct a rigid bounded operator of $l^2$ the spectrum of which is $\{λ\in \mathbb C: r \leq | λ| \leq 1\}$. For $0 < r < 1$, it gives the first examples of rigid bounded invertible operators, such that their inverse is not rigid.

math.FA↗
arXiv2018

On path partitions of the divisor graph

It is known that the longest simple path in the divisor graph that uses integers $\leq N$ is of length $\asymp N/\log N$. We study the partitions of $\{1,2,\dots, N\}$ into a minimal number of paths of the divisor graph, and we show that in such a partition, the longest path can have length asymptotically $N^{1-o(1)}$.

math.NT↗
arXiv2018

Etude du graphe divisoriel 4

We show that there is a permutation $f$ of the positive integers such that for $n \geq 2,$ l.c.m.$(f(n), f(n+1)) \leq cn(\log n)^2,$ where $c$ is a positive constant. It improves previous results of Erdös, Freud and Hegyvari (1983), and Chen and Ji (2011).

math.NT↗
arXiv2015

Fonctions complètement multiplicatives de somme nulle

Completely multiplicative functions whose sum is zero ($CMO$).The paper deals with $CMO$, meaning completely multiplicative ($CM$) functions $f$ such that $f(1)=1$ and $\sum\limits\_1^\infty f(n)=0$. $CM$ means $f(ab)=f(a)f(b)$ for all $(a,b)\in \N^{*2}$, therefore $f$ is well defined by the $f(p)$, $p$ prime. Assuming that $f$ is $CM$, give conditions on the $f(p)$, either necessary or sufficient, both is possible, for $f$ being $CMO$ : that is the general purpose of the authors.The $CMO$ character of $f$ is invariant under slight modifications of the sequence $(f(p))$ (theorem 3). The same idea applies also in a more general context (theorem 4).After general statements of that sort, including examples of $CMO$ (theorem 5), the paper is devoted to "small" functions, that is, functions of the form $\frac{f(n)}{n}$, where the $f(n)$ are bounded. Here is a typical result : if $|f(p)|\le 1$ and $Re\, f(p)\le0$ for all $p$, a necessary and sufficient condition for $\big(\frac{f(n)}{n}\big)$ to be $CMO$ is $\sum \, Re\, f(p)/p=-\infty$ (theorem 8). Another necessary and sufficient condition is given under the assumption that $|1+f(p)|\le 1$ and $f(2)\not=-2$ (theorem 7). A third result gives only a sufficient condition (theorem 9). The three results apply to the particular case $f(p)=-1$, the historical example of Euler.Theorems 7 and 8 need auxiliary results, coming either from the existing literature (Halász, Montgomery--Vaughan), or from improved versions of classical results (Ingham, Skał ba) about $f(n)$ under assumptions on the $f*1(n)$, * denoting the multiplicative convolution (theorems 10 and 11).

math.NT↗
arXiv2008

Zeros of Dirichlet series with periodic coefficients

Let $a=(a_n)_{n\ge 1}$ be a periodic sequence, $F_a(s)$ the meromorphic continuation of $\sum_{n\ge 1} a_n/n^s$, and $N_a(σ_1, σ_2, T)$ the number of zeros of $F_a(s)$, counted with their multiplicities, in the rectangle $σ_1 < \Re s < σ_2$, $|\Im s | \le T$. We extend previous results of Laurinčikas, Kaczorowski, Kulas, and Steuding, by showing that if $F_a(s)$ is not of the form $P(s) L_χ (s)$, where $P(s)$ is a Dirichlet polynomial and $L_χ(s)$ a Dirichlet L-function, then there exists an $η=η(a)>0$ such that for all $1/2 < σ_1 < σ_2 < 1+η$, we have $c_1 T \le N_a(σ_1, σ_2, T) \le c_2 T$ for sufficiently large $T$, and suitable positive constants $c_1$ and $c_2$ depending on $a$, $σ_1$, and $σ_2$.

math.NT↗