arXiv subjects

Armand Lachand

Publications and source records attributed to Armand Lachand.

3 recordsLinked to original sources

arXiv2017

On the representation of friable integers by linear forms

Let $P^+(n)$ denote the largest prime of the integer $n$. Using the \begin{align*}Ψ\_{F\_1\cdots F\_t}\left(\mathcal{K}\cap[-N,N]^d,N^{1/u}\right):=\\#\left\{\mathcal{K}\in {\mathbf{N}}\cap[-N,N]^d:\vphantom{P^+(F\_1(\boldsymbol{n})\cdots F\_t(\boldsymbol{n}))\leq N^{1/u}}\right.\left.P^+(F\_1(\boldsymbol{n})\cdots F\_t(\boldsymbol{n}))\leq N^{1/u}\right\}\end{align*} where $(F\_1,\ldots,F\_t)$ is a system of affine-linear forms of $\mathbf{Z}[X\_1,\ldots,X\_d]$ no two of which are affinely related and $\mathcal{K}$ is a convex body. This improves upon Balog, Blomer, Dartyge and Tenenbaum's work~\cite{BBDT12} in the case of product of linear forms.

math.NT↗

arXiv2014

Fonctions arithmétiques et formes binaires irréductibles de degré $3$

Let $F(X_1,X_2)\in\mathbb{Z}[X_1,X_2] $ be an irreducible binary form of degree $3$ and $h$ an arithmetic function. We give some estimates for the average order $\sum_{\substack{|n_1|\leq x,|n_2|\leq x}}h(F(n_1,n_2))$ when $h$ satisfy certain conditions. As an application, we provide some asymptotic formula for the number of $y$-friable values of $F(n_1,n_2)$ when the variables $n_1,n_2$ lies in the square $[1,x]^2$ and uniformly in the region $\exp\left(\frac{\log x}{(\log\log x)^{1/2-\varepsilon}}\right)\leq y\leq x$. This improves a result of Balog, Blomer, Dartyge and Tenenbaum (2012).

math.NT↗

arXiv2014

Some mathematical remarks on the polynomial selection in NFS

In this work, we consider the proportion of smooth (free of large prime factors) values of a binary form $F(X_1,X_2)\in\Z[X_1,X_2]$. In a particular case, we give an asymptotic equivalent for this proportion which depends on $F$. This is related to Murphy's $α$ function, which is known in the cryptographic community, but which has not been studied before from a mathematical point of view. Our result proves that, when $α(F)$ is small, $F$ has a high proportion of smooth values. This has consequences on the first step, called polynomial selection, of the Number Field Sieve, the fastest algorithm of integer factorization.

cs.CR↗