arXiv subjects
Publications and source records attributed to Régis de la Bretèche.
For integer $n\geqslant 1$ and real number $z\geqslant 1$, define $M(n,z):=\sum_{d|n,\,d\leqslant z}μ(d)$ where $μ$ denotes the Möbius function. Put ${\cal L}(y):=\exp\left\{(\log y)^{3/5}/(\log_2y)^{1/5}\right\}$ $(y\geqslant 3)$. We show that, for a suitable, explicit, constant $L>0$ and some absolute $c>0$, we have $S(x,z)= Lx+O\left({x/{\cal L}(3ξ)^c}\right)$ uniformly for $x\geqslant 1$, $ξ\leqslant z\leqslant x/ξ$.
In recent years the question of maximizing GCD sums regained interest due to its firm link with large values of $L$-functions. In the present paper we initiate the study of minimizing for positive weights~$w$ of normalized $L^1$- norm the sum $\sum_{m_1 , m_2 \leqslant N} w({m_1})w({m_2})\frac{(m_1,m_2)}{\sqrt{m_1m_2}} $. We consider as well the intertwined question of minimizing a weighted version of the usual multiplicative energy. We give three applications of our results. Firstly we obtain a logarithmic refinement of Burgess' bound on character sums $\displaystyle{\sum_{M 0$, there exists at least $ \gg p/(\log p)^{ δ+o(1)}$ (with $δ=1-\frac{1+\log_2 2}{\log 2} \approx 0.08607$) even characters such that $θ(x,χ) \neq 0$. Lastly we obtain lower bounds on small moments of character sums.
We obtain new estimates on the level of distribution of the set $\{Q(n)\}$ where $Q\in{\mathbb Z}[X]$ is irreducible quadratic, for well-factorable moduli, improving a result due to Iwaniec. As a by-product of our arguments, we study the Chebyshev problem of estimating $\max\{P^+(n^2-D), n\leq x\}$ and make explicit, in Deshouillers-Iwaniec's state-of-the-art result, the dependence on the Selberg eigenvalue conjecture. Combined with the construction of an upper-bound sieve for numbers free of large factors, we obtain new upper bounds for the quantity $Ψ_Q(x, y) = |\{n\leq x: p\mid Q(n)\Rightarrow p\leq y\}|$ for $Q\in{\mathbb Z}[X]$ linear or quadratic.
Let $s_a(n)$ denote the sum of digits of an integer $n$ in the base $a$ expansion. Answering, in a extended form, a question of Deshouillers, Habsieger, Laishram, and Landreau, we show that, provided $a$ and $b$ are multiplicatively independent, any positive real number is a limit point of the sequence $\{s_b(n)/s_a(n)\}_{n=1}^{\infty}$. We also provide upper and lower bounds for the counting functions of the corresponding subsequences.
In this note, we establish an asymptotic formula for the number of rational points of bounded height on the singular cubic surface $$ x_0(x_1^2 + x_2^2)=x_3^3 $$ with a power-saving error term, which verifies the Manin-Peyre conjectures for this surface.
We obtain new results concerning the Sato-Tate conjecture on the distribution of Frobenius traces over single and double parametric families of elliptic curves. We consider these curves for values of parameters having prescribed arithmetic structure: product sets, geometric progressions, and most significantly prime numbers. In particular, some families are much thinner than the ones previously studied.
We investigate Sarnak's conjecture on the Möbius function in the special case when the test function is the indicator of the set of integers for which a real additive function assumes a given value.
Elaborating on previous works and taking advantage of estimates on the local behaviour of the counting function of friable integers, we determine the optimal range in which the friable Turán-Kubilius constant tends to $1$.
We give estimates for the first two moments of arithmetical sequences in progressions. Instead of using the standard approximation, we work with a generalization of Vaughan's major arcs approximation which is similar to that appearing in earlier work of Browning and Heath-Brown on norm forms. We apply our results to the sequence $τ_k(n)$, and obtain unconditional results in a wide range of moduli.
We study the average error term in the usual approximation to the number of $y$-friable integers congruent to $a$ modulo $q$, where $a\neq 0$ is a fixed integer. We show that in the range $\exp\{(\log\log x)^{5/3+\varepsilon}\} \leq y \leq x$ and on average over $q\leq x/M$ with $M\rightarrow \infty$ of moderate size, this average error term is asymptotic to $-|a|Ψ(x/|a|,y)/2x$. Previous results of this sort were obtained by the second author for reasonably dense sequences, however the sequence of $y$-friable integers studied in the current paper is thin, and required the use of different techniques, which are specific to friable integers.
We count rational points of bounded height on the Cayley ruled cubic surface and interpret the result in the context of general conjectures due to Batyrev and Tschinkel.