On a logarithmic sum related to a natural quadratic sieve

We study the sum $Σ_q(U)=\sum_{\substack{d,e\leq U\\(de,q)=1}}\frac{μ(d)μ(e)}{[d,e]}\log\left(\frac{U}{d}\right)\log\left(\frac{U}{e}\right)$, $U>1$, 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 $Σ_1(U)$ is asymptotic to $\log(U)$. In this article, we discover not only that $Σ_q(U)\sim\frac{q}{φ(q)}\log(U)$ for all $q\in\mathbb{Z}_{>0}$, but also we find a closed-form expression for its secondary order term of $Σ_q(U)$, a constant $\mathfrak{s}_q$, which we are able to estimate explicitly when $q=v\in\{1,2\}$. We thus have $Σ_v(U)= \frac{v}{φ(v)}\log(U)-\mathfrak{s}_v+O_v^*\left(\frac{K_v}{\log(U)}\right)$, 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.

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