EN
Let λ(n) be the Liouville function. We find a nontrivial upper bound for the sum $$ \sum\limits_{X \leqslant n \leqslant 2X} {\lambda (n)e^{2\pi i\alpha \sqrt n } } ,0 \ne \alpha \in \mathbb{R} $$ The main tool we use is Vaughan’s identity for λ(n).