On a logarithmic sum related to the Selberg sieve
Tom 202 / 2022
Streszczenie
We study the sum $$\Sigma _q(U)=\sum _{\substack {d,e\leq U\\(de,q)=1}}\frac {\mu (d)\mu (e)}{[d,e]} \log \biggl (\frac {U}{d}\bigg )\log \biggl (\frac {U}{e}\bigg ),\quad U \gt 1,$$ so that a continuous, monotonic and explicit version of Selberg’s sieve can be stated.
Thanks to Barban–Vekhov (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 $q\in \mathbb {Z}_{ \gt 0}$, but also we find a closed-form expression for the second 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 have $\Sigma _v(U)= \frac {v}{\varphi (v)}\log (U)-\mathfrak {s}_v+O^*\bigl (\frac {K_v}{\log (U)}\big )$, for some explicit constant $K_v \gt 0$, 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.
