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 \(|\alpha -a/q| \le 1/q^2\) with \((a,q)=1\), then for any \(\varepsilon >0\),

$$\begin{aligned} \qquad \qquad \sum _{n\leqslant X}{\lambda _f(n)}^2 e(n\alpha ) \ll _{f, \varepsilon } X^{{\frac{4}{5}}+\varepsilon } \quad \text {for} \ X^{{\frac{1}{5}}} \ll q \ll X^{{\frac{4}{5}}}. \end{aligned}$$

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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The main part of this work was done during the author’s Ph.D. thesis at the Indian Statistical Institute (ISI), Kolkata where he was a student. The author thanks Satadal Ganguly, Stephan Baier and Ritabrata Munshi for helpful suggestions. The author also thanks ISI for financial support and excellent working atmosphere.

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Correspondence to Ratnadeep Acharya.

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Communicating Editor: A Raghuram

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Acharya, R. Exponential sums of squares of Fourier coefficients of cusp forms. Proc Math Sci 130, 24 (2020). https://doi.org/10.1007/s12044-019-0550-4

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  • Cusp forms
  • exponential sums
  • diophantine approximation

Mathematics Subject Classification

  • 11F30