  Let $\lambda$ the Barban--Vehov weights, defined in $(1)$. Let $X\ge
z_1\ge100$ and $z_2=z_1^\tau$ for some $\tau>1$. We prove that
  \begin{equation*}
  \sum_{n\le
  X}\frac{1}{n}\Bigl(\sum_{\substack{d|n}}\lambda_d\Bigr)^2
  \le
  f(\tau)\frac{\log X}{\log (z_2/z_1)},
  \end{equation*}
  for a completely determined function $f:(1,\infty)\to\mathbb{R}_{>0}$.
  In particular, we may take $f(2)=30$, saving more than a factor of $5$ on
what was the best known result for $\tau=2$. Two related estimates are also
provided for general $\tau>1$.
