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 ${\mathrm{\Sigma }}_1(U)$ is asymptotic to $\log (U)$. In this article, we discover not only that ${\mathrm{\Sigma }}_q(U)\sim \frac{q}{\phi (q)}\log (U)$ for all $q\in {\mathbb{Z}}_{>0}$, but also we find a closed-form expression for its secondary order term of ${\mathrm{\Sigma }}_q(U)$, 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 ${\mathfrak{s}}_1=0.60731\dots$ 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.