On an extremal problem. (Russian) Zbl 0167.04003

Barban, M. B.; Vekhov, P. P.

Tr. Mosk. Mat. O.-va 18, 83-90 (1968).

The following theorem is proved: Let \(\lambda_d=\mu(d)\) when \(d\leq z\) and \(=\mu(d)(\log z_1/d)(\log z_1/z)^{-1}\) when \(z\leq d\leq z_1\). Then for \(x>z\), \(\log z_1\ll \log z\) we have \[ \sum_{1<n\leq x}\left(\sum_{d\mid n,\;d\leq z_1}\lambda_d\right)^2\ll x\left(\log\frac{z_1}{z}\right)^{-1}. \]

Reviewer: Imre Kátai (Budapest)

MSC:

11N37	Asymptotic results on arithmetic functions
11N35	Sieves

Keywords:

estimate of arithmetic function