zbMATH — the first resource for mathematics

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.

11L07 Estimates on exponential sums
11F11 Holomorphic modular forms of integral weight
11F30 Fourier coefficients of automorphic forms
Full Text: DOI Euclid arXiv
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.