Abstract:
The class $\mathcal{F}$ consisting of all multiplicative functions $f$ satisfying the inequality $|f(p)|\leq A$ for some constant $A\geq 1$ and all primes $p$ and $\sum_{n=1}^N |f(n)|^2\leq A^2N$ is considered. It is proved that for any real irrational algebraic $\alpha$ and for all natural numbers $k$ and $N$ the following estimate holds uniformly over all multiplicative functions $f$ from $\mathcal{F}$: $$ S(\alpha)=\sum_{n=1}^Nf(n)\rho(n\alpha)\ll_A\frac{N}{\ln N}, $$ where $\rho(t)=0,5-\{t\}.$
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