In 1968, Barban and Vehov considered [1] the problem of determining for which continuous functions $\rho:\mathbb{R}^+\to [0,1]$ satisfying certain properties ($\rho(t)=1$ for $t\leq U_0$, $\rho(t)=0$ for $t>U_1$) the sum $$S(x) = \sum_{n\leq x} \left(\mathop{\sum_{d|n}}_{d\leq U_0} \rho(d) \mu(d) \right)^2$$ was minimal. (Assume from now on that $x>U_1>U_0$. They showed that the choice $\rho(t) = \rho_0(t) = \log(t/U_0)/\log(U_1/U_0)$ for $U_0<t\leq U_1$ was grosso modo optimal, in the sense of giving a sum $S(x)$ whose main term is no more than a constant times the minimal value that such a sum $S(x)$ could take. Later, Graham showed that, for $\rho(t) = \rho_0(t)$ as above, $S(x)$ actually asymptotes to $x/\log(U_1/U_0)$.
(a) What is the easiest way to show that this is the least possible leading term in an asymptotic for $S(x)$ for any function $\rho$ satisfying the above conditions? (That the order of magnitude is correct is clear from estimates on rough numbers, but such estimates have Buchstab's function in front.)
(b) What work, if any, has been done to date on the second-order term in the asymptotics? Graham gave a bound of the form $x/\log(U_1/U_0) + O(x/\log^2(U_1/U_0))$. I recently finished showing that, for $\rho$ as above and $U_0\ggg \sqrt{X}$, $S(x)$ asymptotes to $x/\log(U_1/U_0) - c x/\log^2(U_1/U_0)$, where $c$ is an explicit constant that is positive under some broad conditions ($U_1/U_0\geq 8$). I have no idea as to whether $c$ is the best value one could obtain for any $\rho$ satisfying the conditions.
In the range $U_1\lll X$, one could of course compare the corresponding constant $c'$ to the constant coming from Selberg's sieve. At the same time, one could change the conditions slightly (requiring $\rho$ to be the rescaling of a continuous function $\eta$ independent of $U_0$, $U_1$ and $X$, say) so as to exclude Selberg's sieve from consideration -- and then we would only have an upper bound on $c$. Is anything better known?
[1] M. B. BARBAN AND P. P. VEHOV, On an extremal problem, Trudy Moscov. Obsch. 18 (1968), 83-90.
