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The generalized Bourgain-Sarnak-Ziegler criterion and its application to additively twisted sums on \(\mathrm{GL}_m\). (English) Zbl 1483.11212

Suppose that \(\alpha\) be a real number and \(e(n\alpha)=\exp\{2\pi\textbf{i}n\alpha\}\) for natural numbers \(n\). The authors of the paper derive an upper bound for the sum \[ \sum_{n\leqslant x}a(n)e(n\alpha), \] where \(a\) be multiplicative arithmetic function satisfying certain requirements.
Using deeper methods, the authors get a non-trivial uniform estimates for sums \[ \sum_{n\leqslant x}a(n)e(n^k\alpha),\ \ \sum_{n\leqslant x}a(n)\mu(n)e(n^k\alpha), \] where \(k\geqslant 1\), \(\mu\) is the Möbius function, and \(\{a(n)\}, n\in\mathbb{N},\) are the Dirichlet coefficients of a certain \(L\)-function. The special attention is paid to \(L\)-functions of automorphic cups forms on \(\mathrm{GL}_m\) over \(\mathbb{Q}\) with \(m\geqslant 2\).

MSC:

11N37 Asymptotic results on arithmetic functions
11L07 Estimates on exponential sums
11F30 Fourier coefficients of automorphic forms
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
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