RATNADEEP ACHARYA
Articles written in Proceedings – Mathematical Sciences
Volume 130 All articles Published: 12 March 2020 Article ID 0024 Research Article
Exponential sums of squares of Fourier coefficients of cusp forms
We prove nontrivial estimates for linear sums of squares of Fourier coefficients of holomorphic and Maass cusp forms twisted by additive characters. For holomorphic forms $f$ , we show that if $|\alpha − a/q| \leq 1/q^{2}$ with $(a, q) = 1$, then for any $\varepsilon$ > 0,
$$\sum_{n\leqslant x}\lambda_{f}(n)^{2}e(n\alpha)\, \ll _{f,\varepsilon} X^{\frac{4}{5}+\varepsilon}\,\, for X^\frac{1}{5}\, \ll \,q \, \ll \, X^\frac{4}{5}.$$
Moreover, for any $\varepsilon$ > 0, there exists a set $S \subset (0, 1)$ with $\mu(S) = 1$ such that for every $\alpha \in S$, there exists $X_{0} = X_{0}(\alpha)$ such that the above inequality holds true for any $\alpha \in S$ and $X \geqslant X_{0}(\alpha)$. A weaker bound for Maass cusp forms is also established.
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