Eric Saias

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 $\{1,\dots , N\}$ into a \emph {minimal} number of paths in the divisor graph, and we show that in such

We study multiplicative functions $f$ satisfying $|f(n)|\le 1$ for all $n$, the associated Dirichlet series $F(s):=\sum_{n=1}^{\infty} f(n) n^{-s}$, and the summatory function $S_f(x):=\sum_{n\le x} f(n)$. Up to a possible trivial contribution from the numbers $f(2^k)$, $F(s)$ may have at most one zero or one pole on the one-line, in a sense made precise by Hal\'{a}sz. We estimate $\log F(s)$ away from any such point and show that if $F(s)$ has a zero on the one-line in the sense of Hal\'{a}sz, then $|S_f(x)|\le (x/\log x) \exp\big(c\sqrt{\log \log x}\big)$ for all $c>0$ when $x$ is large enough. This bound is best possible.

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 Erd\"os, Freud and Hegyvari (1983), and Chen and Ji (2011).

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 (Hal\'asz, Montgomery--Vaughan), or from improved versions of classical results (Ingham, Ska{\l} ba) about $f(n)$ under assumptions on the $f*1(n)$, * denoting the multiplicative convolution (theorems 10 and 11).

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 $N_a(\sigma_1, \sigma_2, T)$ the number of zeros of $F_a(s)$, counted with their multiplicities, in the rectangle $\sigma_1 < \Re s < \sigma_2$, $|\Im s | \le T$. We extend previous results of Laurin\v{c}ikas, Kaczorowski, Kulas, and Steuding, by showing that if $F_a(s)$ is not of the form $P(s) L_{\chi} (s)$, where $P(s)$ is a Dirichlet polynomial and $L_{\chi}(s)$ a Dirichlet L-function, then there exists an $\eta=\eta(a)>0$ such that for all $1/2 < \sigma_1 < \sigma_2 < 1+\eta$, we have $c_1 T \le N_a(\sigma_1, \sigma_2, T) \le c_2 T$ for sufficiently large $T$, and suitable positive constants $c_1$ and $c_2$ depending on $a$, $\sigma_1$, and $\sigma_2$.