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On the mean square of short exponential sums related to cusp forms. (English) Zbl 1243.11084
Let $$f(z)=\sum_{n\geq 1}a(n)n^{(\kappa -1)/2} e(nz)$$ be a holomorphic cusp form of weight $$\kappa$$ with respect to the full modular group. The length exponential sum is defined by $\sum\limits_{M\leq n \leq M+\Delta}a(n)e(n\alpha),$ where $$\alpha$$ is a real number. In this paper, the authors estimate the mean square of a square-root length exponential sum of Fourier a holomorphic cusp form, that is the following result $\int\limits_{M}^{M+\Delta}\bigg|\sum\limits_{x \leq n \leq x+\sqrt[]{x}}a(n)e(\frac{hn}{k})\bigg|^{2}\omega(x)\,dx\ll \Delta M^{1/2+\varepsilon},$ where the constant implied by the $$\ll$$ symbol depends only on $$\varepsilon$$, and $$h$$ and $$k$$ be integers with $$0\leq h<k\ll M^{1/2}$$, furthermore, $$\omega(x)$$ denotes a smooth weight function that is supported on the interval $$[M, M+\Delta]$$ where $$k^{2}M^{1/2+\delta} \ll \Delta \ll M$$ with $$\delta$$ an arbitrarily small fixed positive real number.

##### MSC:
 11L07 Estimates on exponential sums 11F11 Holomorphic modular forms of integral weight 11F30 Fourier coefficients of automorphic forms
##### Keywords:
exponential sums; Fourier coefficients; mean square
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##### References:
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