an:05778235
Zbl 1211.11095
Ernvall-Hyt??nen, Anne-Maria
A note concerning certain exponential sums related to cusp forms
EN
??iauliai Math. Semin. 4(12), 75-82 (2009).
00254202
2009
j
11L07 11F11 11F30
divisor function; estimates; exponential sums; Fourier series; holomorphic cusp forms; Maass forms
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.
Huaning Liu (Shaanxi)