An $L^2$-bound for the Barban-Vehov weights

Let $\lambda$ the Barban–Vehov weights, defined in $(1)$. Let $X\ge z_1\ge 100$ 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}}λ_d\Bigr)^2 \le f(τ)\frac{\log X}{\log (z_2/z_1)}, \end{equation*} for a completely determined function $f:(1,\mathrm{\infty })\to {\mathbb{R}}_{>0}$. 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 $\tau >1$.