On the friable Turán-Kubilius inequality (Q2793765)

The authors of this paper present two new forms of the Turan-Kubilius inequality for friable numbers together with several applications of this inequality. Special attention is intended to the large sieve inequality which can be obtained using the principle of duality. The first main assertion of the paper states that NEWLINE\[NEWLINE\begin{aligned} \frac{1}{\#S(x,y)}\sum\limits_{n\in S(x,y)}\Big|f(n)-\sum\limits_{p^\nu\in S(x,y)}\frac{g_p(\alpha)f(p^\nu)}{p^{\nu\alpha}}\Big|^2\\ \ll\sum\limits_{p^\nu\in S(x,y)}\frac{g_p(\alpha)}{p^{\nu\alpha}}{|f(p^\nu)|^2}-\sum\limits_{p\leqslant y}\Big|\sum\limits_{1\leqslant\nu\leqslant\nu_p}\frac{g_p(\alpha)f(p^\nu)}{p^{\nu\alpha}}\Big|^2 \end{aligned}NEWLINE\]NEWLINE uniformly for all \(2\leqslant y\leqslant x\) and for all additive complex arithmetic functions \(f\). Here \(S(x,y)\) is the set of \(y\)-friable integer numbers not exceeding \(x\); \(\nu_p=\nu_p(x)=\lfloor \log x/\log p\rfloor\); \(g_p(\alpha)=1-1/p^\alpha\) and \(\alpha=\alpha(x,y)\) is the solution of the equation NEWLINE\[NEWLINE \sum\limits_{p\leqslant y}\frac{\log p}{p^\alpha-1}=\log x. NEWLINE\]