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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
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[1] A.-M. Ernvall-Hytönen, A relation between Fourier coefficients of holomorphic cusp forms and exponential sums , Publications de l’Institut Mathematique (Beograd), 86 (100) (2009), 97-105.
[2] A.-M. Ernvall-Hytönen and A. Lepistö, Bounds and computational results for exponential sums related to cusp forms , Acta Mathematica Universitatis Ostraviensis 17 (2009), 81-90. · Zbl 1180.90246
[3] A.-M. Ernvall-Hytönen and K. Karppinen, On short exponential sums involving Fourier coefficients of holomorphic cusp forms , Int. Math. Res. Not. IMRN, (10):Art. ID. rnn022, 44 , 2008. · Zbl 1247.11106
[4] A.-M. Ernvall-Hytönen, An improvement on the upper bound of exponential sums of holomorphic cusp forms , submitted.
[5] A. Ivić, On the divisor function and the Riemann zeta-function in short intervals , Ramanujan J. 19 (2) (2209), 207-224. · Zbl 1226.11086 · doi:10.1007/s11139-008-9142-0
[6] A. Ivić, On the mean square of the divisor function in short intervals , J. Théor. Nombres Bordeaux 21 (2) (2009), 251-261. · Zbl 1222.11115 · doi:10.5802/jtnb.669 · numdam:JTNB_2009__21_2_251_0 · eudml:10879
[7] M. Jutila, On exponential sums involving the divisor function , J. Reine Angew. Math. 355 (1985), 173-190. · Zbl 0542.10032 · doi:10.1515/crll.1985.355.173 · crelle:GDZPPN002202115 · eudml:152693
[8] M. Jutila, Lectures on a Method in the Theory of Exponential Sums , volume 80 of Tata Institute of Fundamental Research Lectures on Mathematics and Physics, Published for the Tata Institute of Fundamental Research, Bombay, 1987. · Zbl 0671.10031
[9] M. Jutila, On exponential sums involving the Ramanujan function , Proc. Indian Acad. Sci. Math. Sci. 97 (1-3) (1987), 157-166 (1988). · Zbl 0658.10043 · doi:10.1007/BF02837820
[10] M. Jutila and Y. Motohashi, Uniform bound for Hecke \(L\) -functions, Acta Math. 195 (2005), 61-115. · Zbl 1098.11034 · doi:10.1007/BF02588051
[11] J. R. Wilton, A note on Ramanujan’s arithmetical function \(\tau(n)\) , Proc. Cambridge Philos. Soc. 25 (II) (1929), 121-129. · JFM 55.0709.02
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