Strong orthogonality between the Möbius function, additive characters and Fourier coefficients of cusp forms (Q2877502)

The main result of this nice paper is the bound NEWLINE\[NEWLINE\sum_{n \leq X} \mu(n) \nu_f(n) e(\alpha n) \ll_f X \exp(- c_0 \sqrt{\log X})NEWLINE\]NEWLINE for \(\alpha \in \mathbb{R}\), \(X \geq 2\) and some constant \(c_0\), where \(\nu_f\) is the sequence of the Fourier coefficients of any cuspidal automorphic form for \(\text{SL}_2(\mathbb{Z})\). A similar bound is obtained where the Möbius function is replaced with the characteristic function on the primes.NEWLINENEWLINEThis can be seen as a variation of Wilton's bound with the Möbius function included or as a \(\text{GL}(2)\) prime number theorem with arbitrary additive twists (the corresponding \(\text{GL}(1)\) result without the factor \(\nu_f(n)\) is due to Davenport). Finally it can be interpreted as a proof of the Möbius randomness law for the function \(n \mapsto \nu_f(n) e(\alpha n)\).NEWLINENEWLINEThe proof uses two different methods. For well-approximable \(\alpha\), the twist \(e(\alpha n)\) can be replaced by a multiplicative twist \(\chi(n)\) (with a small error) with a Dirichlet character \(\chi\) to small modulus. In this case, the authors prove, by classical techniques, a \(\text{GL}(2)\) prime number theorem in residue classes. For badly approximable \(\alpha\), the authors use a combinatorial device to replace the Möbius function, often known as Vaughan's identities. The hard part is the treatment of the type II sums, where sums over \(\lambda_f(n^2) e(\alpha n)\) need to be bounded uniformly in \(\alpha\), which follows from a result of \textit{S. D. Miller} [Am. J. Math. 128, No. 3, 699--729 (2006; Zbl 1142.11033)].