On the estimation of nonlinear twists of the Liouville function (Q384325)

The author has provided a nice proof of a nontrivial upper bound for the expression NEWLINE\[NEWLINE\Biggl|\sum_{X\leq n\leq 2X} \lambda(n)\,e(\alpha\sqrt{n})\Biggr|,NEWLINE\]NEWLINE where \(\alpha\in\mathbb R\setminus\{0\}\), and as usual \(e(n)= e^{2\pi in}\). Earlier upper bounds for the summation NEWLINE\[NEWLINES(X,\alpha)= \sum_{X\leq n\leq 2X} a_n e(\alpha\sqrt{n}),\quad \alpha\in\mathbb R\{0\}\tag{1}NEWLINE\]NEWLINE have been proved by a number of mathematicians, starting with \textit{I. M. Vinogradov}, who studied in [``Special variants of the method of trigonometric sums'' (Russian). Moskva: Izdatel'stvo ``Nauka'' (1976; Zbl 0429.10023)] the sum (1) in the case when \(a_n\) is the von Mangoldt function \(\Lambda(n)\).NEWLINENEWLINE In [Publ. Math., Inst. Hautes Étud. Sci. 91, 55--131 (2000; Zbl 1012.11041)] \textit{H. Iwaniec} et al. proved that in the case when \(a_n\) is either \(\Lambda(n)\) or the Möbius function \(\mu(n)\), the sums of type (1) are very much related to the \(L\)-functions of \(\mathrm{GL}_2\).