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

TL;DR

This paper establishes an $L^2$-bound for the Barban-Vehov weights, providing explicit bounds and improvements over previous results, with applications to number theory and related estimates.

Contribution

The paper derives a new explicit $L^2$-bound for Barban-Vehov weights, improving the known bounds for the case $ au=2$ and extending estimates to general $ au>1$.

Findings

Proves an explicit bound for the sum involving Barban-Vehov weights.

Achieves a factor of over 5 improvement for $ au=2$ case.

Provides related estimates for all $ au>1$.

Abstract

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 $\tau>1$.