Explicit averages of square-free supported functions: to the edge of the convolution method

We give a general statement of the convolution method so that one can provide explicit asymptotic estimations for all averages of square-free supported arithmetic functions that have a sufficiently regular order on the prime numbers and observe how the nature of this method gives error term estimations of order $X^{-\delta}$, where $\delta$ belongs to an open real positive set $I$. In order to have a better error estimation, a natural question is whether or not we can achieve an error term of critical order $X^{-\delta_0}$, where $\delta_0$, the critical exponent, is the right hand endpoint of $I$. We reply positively to that question by presenting a new method that improves qualitatively almost all instances of the convolution method under some regularity conditions; now, the asymptotic estimation of averages of well-behaved square-free supported arithmetic functions can be given with its critical exponent and a reasonable explicit error constant. We illustrate this new method by analyzing a particular average related to the work of Ramar\'e--Akhilesh (2017), which leads to notable improvements when imposing non-trivial coprimality conditions.Comment: Updated. Some correction

On a logarithmic sum related to a natural quadratic sieve

We study the sum $\Sigma_q(U)=\sum_{\substack{d,e\leq U\\(de,q)=1}}\frac{\mu(d)\mu(e)}{[d,e]}\log\left(\frac{U}{d}\right)\log\left(\frac{U}{e}\right)$, $U>1$, so that a continuous, monotonic and explicit version of Selberg's sieve can be stated. Thanks to Barban-Vehov (1968), Motohashi (1974) and Graham (1978), it has been long known, but never explicitly, that $\Sigma_1(U)$ is asymptotic to $\log(U)$. In this article, we discover not only that $\Sigma_q(U)\sim\frac{q}{\varphi(q)}\log(U)$ for all $q\in\mathbb{Z}_{>0}$, but also we find a closed-form expression for its secondary order term of $\Sigma_q(U)$, a constant $\mathfrak{s}_q$, which we are able to estimate explicitly when $q=v\in\{1,2\}$. We thus have $\Sigma_v(U)= \frac{v}{\varphi(v)}\log(U)-\mathfrak{s}_v+O_v^*\left(\frac{K_v}{\log(U)}\right)$, for some explicit constant $K_v > 0$, where $\mathfrak{s}_1=0.60731\ldots$ and $\mathfrak{s}_2=1.4728\ldots$. As an application, we show how our result gives an explicit version of the Brun-Titchmarsh theorem within a range.Comment: accepted in Acta Arithmetic

Dynamics of B\mathscr{B}-free systems generated by Behrend sets. I

We study the complexity of $\mathscr{B}$-free subshifts which are proximal and of zero entropy. Such subshifts are generated by Behrend sets. The complexity is shown to achieve any subexponential growth and is estimated for some classical subshifts (prime and semiprime subshifts). We also show that $\mathscr{B}$-admissible subshifts are transitive only for coprime sets $\mathscr{B}$ which allows one to characterize dynamically the subshifts generated by the Erd\"os sets

Explicit L2L^2 bounds for the Riemann ζ\zeta function

Bounds on the tails of the zeta function $\zeta$, and in particular explicit bounds, are needed for applications, notably for integrals involving $\zeta$ on vertical lines or other paths going to infinity. An explicit version of the traditional `convexity bound' has long been known (Backlund 1918). To do better, one must either provide explicit versions of subconvexity bounds, or give explicit bounds on means of $\zeta$. Here we take the second road, bounding weighted $L^2$ norms of tails of $\zeta$. Two approaches are followed, each giving the better result on a different range. One of them is inspired by the proof of the standard mean value theorem for Dirichlet polynomials (Montgomery 1971). The main technical idea is the use of a carefully chosen smooth approximation to $1_{\lbrack 0,1\rbrack}$ so as to eliminate off-diagonal terms. The second approach, superior for large $T$, is based on classical lines, starting with an approximation to $\zeta$ via Euler-Maclaurin. Both bounds give main terms of the correct order for $0<\sigma\leq 1$ and are strong enough to be of practical use in giving precise values for integrals when combined with (rigorous) numerical integration. We also present bounds for the $L^{2}$ norm of $\zeta$ in $[1,T]$ for $0\leq\sigma\leq 1$.Comment: 44 pages; v6: corrections, improvements and clarifications, added reference

Lissage et compensation : une version explicite du crible de Barban-Vehov

Considérons la somme du type crible de Selberg ∑m≤X(∑d|m μ(d)·ρd)2, où d→ρd est un poids pour la somme ∑d|nμ(d). On étudie les poids logarithmiques et ceux de Barban-Vehov, définis respectivement comme L:d→log+(U/d), U>1, V:d→log+(U1/d)−log+(U0/d), 1≤U01, V:d→log+(U1/d)−log+(U0/d), 1≤U0<U1, where log+(x) = max{log(x),0}. V and L were studied by Barban and Vehov (1968) and Motohashi (1974); a first order nonexplicit asymptotic formula for the Selberg sum for those weights was derived in the 70’s byGraham, giving, when using L, a main term identical to the one given by Selberg sieve. The problem of studying explicitly the asymptotic formula of the Selberg sum for L and V andits secondary terms remained open and it was conjectured that those secondary terms were negative, so that we have a negative contribution for the sieve. In this thesis, by using new techniques and some ideas given by H. Helfgott and O. Ramaré,we explicitly obtain, for all coprimality conditions, the second order term of the asymptotic expressions for L and V, which, in the case of interest, v ∈ {1, 2}, confirm their conjectural nature:they are negative. As a closely related object to those weights, we obtain ∑d,e,(de,v)=1 μ(d)μ(e)/[d,e] · LdLe ~ v/φ(v) · log(U) - sv, where s1 = 0.607…, s2 = 1.472… . The fact of having an explicit error term and a negative second order term allows a transition,under mild conditions, from a purely asymptotic result to an actual inequality, finding thus cancellation with respect to the main term. We thus derive an explicit sieve-type result, providingan infinite and wide range of values U0, U1 such that ∑m≤X( ∑d|m μ(d)·Vd / log(U1/U0))2< X /log(U1/U0) - cv · X / log2(U1/U0), for some positive explicitly defined constant c