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

On the computation of singularities of parametrized ruled surfaces

Given a ruled surface V defined in the standard parametric form P(t1, t2), we present an algorithm that determines the singularities (and their multiplicities) of V from the parametrization P. More precisely, from P we construct an auxiliary parametric curve and we show how the problem can be simplified to determine the singularities of this auxiliary curve. Only one univariate resultant has to be computed and no elimination theory techniques are necessary. These results improve some previous algorithms for detecting singularities for the special case of parametric ruled surfaces.Ministerio de Ciencia, Innovación y Universidade