Bounds and computational results for exponential sums related to cusp forms. (English) Zbl 1246.11138

Ernvall-Hytönen, Anne-Maria; Lepistö, Arto

Acta Math. Univ. Ostrav. 17, No. 1, 81-90 (2009).

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 \(M\).

Reviewer: Huaning Liu (Shaanxi)

MSC:

11L07	Estimates on exponential sums
11Y35	Analytic computations

Keywords:

cusp forms; exponential sums; Ramanujan tau function; analytic computations

References:

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