an:05994750
Zbl 1246.11138
Ernvall-Hyt??nen, Anne-Maria; Lepist??, Arto
Bounds and computational results for exponential sums related to cusp forms
EN
Acta Math. Univ. Ostrav. 17, No. 1, 81-90 (2009).
00287242
2009
j
11L07 11Y35
cusp forms; exponential sums; Ramanujan tau function; analytic computations
The holomorphic cusp forms are defined by the Fourier series
\[
F(z)=\sum_{n=1}^{\infty}a(n)n^{\frac{\kappa-1}{2}}\text{e}(nz),
\]
where \(\text{Re} \;z>0\), \(\text{e}(x)=\text{e}^{2\pi i x}\), \(\kappa\) is the weight of the form, and the numbers \(a(n)\) are called normalized Fourier coefficients.
This paper presents some computer data suggesting the size of bounds for exponential sums
\[
\sum_{M\leq n\leq M+\Delta}a(n)\text{e}(n\alpha),
\]
where \(\Delta\) is considerably smaller than \(M\).
Huaning Liu (Shaanxi)