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Oscillations of cusp form coefficients in exponential sums

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Abstract

Let f(z)=∑ ∞n=1 λ(n)n (κ−1)/2 e(nz) be a holomorphic cusp form of weight κ for the full modular group SL 2(ℤ) and let μ(n) be the Möbius function. In this paper, we are concerned with the sum

$$S(\alpha,X)=\sum _{n\leq X}\mu (n)\lambda(n)e(\alpha \sqrt{n}),\quad 0\neq \alpha \in \mathbb{R}.$$

It is proved that, unconditionally, \(S(\alpha,X)\ll X^{\frac{5}{6}}(\log X)^{20}\), where the implied constant depends only on α and the cusp form f.

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Authors and Affiliations

  1. Department of Mathematics, Shandong University, Jinan, 250100, China

    Qinghua Pi & Qingfeng Sun

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  1. Qinghua Pi
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  2. Qingfeng Sun
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Correspondence to Qinghua Pi.

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Pi, Q., Sun, Q. Oscillations of cusp form coefficients in exponential sums. Ramanujan J 21, 53–64 (2010). https://doi.org/10.1007/s11139-008-9151-z

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  • DOI: https://doi.org/10.1007/s11139-008-9151-z

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