On a logarithmic sum related to a natural quadratic sieve

Abstract

We study the sum ${\Sigma }_q(U)={\sum }_{\begin{array}{c}d,e\le U\\ (de,q)=1\end{array}}\frac{\mu (d)\mu (e)}{[d,e]}\log \left(\frac{U}{d}\right)\log \left(\frac{U}{e}\right)$, $U>1$, 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 ${\Sigma }_q(U)\sim \frac{q}{\phi (q)}\log (U)$ 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 ${\Sigma }_v(U)=\frac{v}{\phi (v)}\log (U)-{\mathfrak{s}}_v+O_v^{\ast }\left(\frac{K_v}{\log (U)}\right)$, for some explicit constant $K_v>0$, where ${\mathfrak{s}}_1=0.60731\dots$ and ${\mathfrak{s}}_2=1.4728\dots$. As an application, we show how our result gives an explicit version of the Brun-Titchmarsh theorem within a range.

accepted in Acta Arithmetica

References in corpus (1)

• On the Brun-Titchmarsh Theorem

Cited by in corpus (2)

• From explicit estimates for the primes to explicit estimates for the Möbius function II
• Conversions explicites entre des fonctions sommatoires de la fonction de Möbius