Oscillations of Fourier coefficients of cusp forms over primes (Q746948)

The distribution of Fourier coefficients of modular forms is an important subject in number theory. This problem is studied in the paper under review. Namely, let \(f\) a primitive holomorphic or Maass cusp form for the group \(\mathrm{SL}(2, \mathbb{Z})\), and \(a_f(n)\) its \(n\)th normalised Fourier coefficient. Then in the paper under review, it is shown that for any \(\alpha, \beta \in \mathbb{R}\), there exists an effective positive constant \(c\) such that \[ \sum_{n \leq N} \Lambda(n) a_f(n) e(\alpha n^2 + \beta n ) \ll_f N \exp(-c \sqrt{\log N} ), \] where \(\Lambda\) is the von Mangoldt function and \(N \geq 2\). The proof is based on classical analytic number theory methods.