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

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