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Mean square estimate for relatively short exponential sums involving Fourier coefficients of cusp forms. (English) Zbl 1327.11057
The author estimates the mean square of (relatively) short exponential sums related to the Fourier coefficients of a holomorphic cusp form \(f(z)=\sum_{n=1}^\infty a(n) n^{(\kappa-1)/2} e(nz)\) of weight \(\kappa\), namely, \[ I=\int_{M}^{M+\triangle} w(x) \left|\sum_{x\leq n\leq x+T} a(n) e\left(\frac{h}{k}\, n\right)\right|^2 \; dx. \] In this expression, \(h\) and \(k\) are coprime integers with \(0<k\ll M^{1/2-\varepsilon'}\) for some fixed \(\varepsilon'>0\), \(\triangle=\min(k^2 M^{1/2+\varepsilon}, M)\), \(w(x)\) is a positive pump function on \([M,M+\triangle]\) satisfying certain technical conditions, \(T\asymp M^\delta\) for some \(1/2<\delta<1\) (“relatively short” exponential sum). She obtains \[ I=S+O(k^2 M^{1+\varepsilon}+\triangle M^\varepsilon T^{1/2} k)+O\left(\sqrt{|S|(k^2M^{1+\varepsilon}+\triangle M^\varepsilon T^{1/2}k)}\right). \] The quantity \(S\) is explicit and satisfies \(S\ll \triangle T\) when \(k\gg TM^{-1/2}\) and \(S\ll \triangle M^{1/2}k\) otherwise. The given bounds for \(S\) are the actual orders of magnitude of that term. When omitting the weight function \(w(x)\), she gives upper bounds for \(I\) depending on the size of \(k\). The proof of the general theorem starts off with a Voronoi-type summation formula and proceeds via careful calculations, in particular, to get cancellations on certain diagonal terms.

11L07 Estimates on exponential sums
11F11 Holomorphic modular forms of integral weight
11F30 Fourier coefficients of automorphic forms
Full Text: DOI arXiv
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