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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.

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