Articles written in Proceedings – Mathematical Sciences

    • Exponential sums of squares of Fourier coefficients of cusp forms


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      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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