A note concerning certain exponential sums related to cusp forms. (English) Zbl 1211.11095

Ernvall-Hytönen, Anne-Maria

Šiauliai Math. Semin. 4(12), 75-82 (2009).

The holomorphic cusp forms can be represented as Fourier series \[ F(z)=\sum_{n=1}^{\infty}a(n)n^{\frac{\kappa-1}{2}}e(nz), \] where \(\text{Im}\, z>0\) and the numbers \(a(n)\) are called normalized Fourier coefficients, and \(\kappa\) is the weight of the form. Similarly, the Maass forms can be written as follows \[ u(z)=u(x+iy)=\sqrt{y}\sum_{n\neq 0}t(n)K_{i\kappa}(2\pi |n|y)e(nz) \] with the \(K\)-Bessel functions, where \(\kappa>0\) depends on the eigenvalue of the non-Euclidean Laplacian connected to the form.
This paper considers certain specific exponential sums related to \(a(n)\) and \(t(n)\), and gives some estimates.

Reviewer: Huaning Liu (Shaanxi)

MSC:

11L07	Estimates on exponential sums
11F11	Holomorphic modular forms of integral weight
11F30	Fourier coefficients of automorphic forms

Keywords:

divisor function; estimates; exponential sums; Fourier series; holomorphic cusp forms; Maass forms