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Bounds and computational results for exponential sums related to cusp forms. (English) Zbl 1246.11138
The holomorphic cusp forms are defined by the Fourier series \[ F(z)=\sum_{n=1}^{\infty}a(n)n^{\frac{\kappa-1}{2}}\text{e}(nz), \] where \(\text{Re} \;z>0\), \(\text{e}(x)=\text{e}^{2\pi i x}\), \(\kappa\) is the weight of the form, and the numbers \(a(n)\) are called normalized Fourier coefficients.
This paper presents some computer data suggesting the size of bounds for exponential sums \[ \sum_{M\leq n\leq M+\Delta}a(n)\text{e}(n\alpha), \] where \(\Delta\) is considerably smaller than \(M\).

MSC:
11L07 Estimates on exponential sums
11Y35 Analytic computations
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References:
[1] PARI/GP, Version @vers. 2006. available from · pari.math.u-bordeaux.fr
[2] Apostol, T. M.: Modular functions and Dirichlet series in number theory. volume 41 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1990 · Zbl 0697.10023
[3] Ernvall-Hytönen, A.-M.: A relation between Fourier coefficients of holomorphic cusp forms and exponential sums. to appear in Publications de l’Institut Mathematique · Zbl 1279.11083 · doi:10.2298/PIM0900097E
[4] Ernvall-Hytönen, A.-M., Karppinen, K.: On short exponential sums involving Fourier coefficients of holomorphic cusp forms. Int\?ath. Res. Not. IMRN, (10) : Art. ID. rnn022, 44, 2008 · Zbl 1247.11106 · doi:10.1093/imrn/rnn022
[5] Ernvall-Hytönen, A.-M.: An improvement on the upper bound of exponential sums of holomorphic cusp forms. submitted
[6] Ivić, A.: Large values of certain number-theoretic error terms. Acta Arith., 56(2) : 135-159, 1990 · Zbl 0659.10053 · eudml:206303
[7] Jutila, M.: On exponential sums involving the Ramanujan function. Proc. Indian Acad. Sci. Math. Sci., 97(1-3) : 157-166 (1988), 1987 · Zbl 0658.10043 · doi:10.1007/BF02837820
[8] Koecher, M., Krieg, A.: Elliptische Funktionen und Modulformen. Springer-Verlag, Berlin, 1998 · Zbl 0895.11001
[9] Rankin, R. A.: Contributions to the theory of Ramanujan’s function \(\tau (n)\) and similar arithmetical functions ii. The order of Fourier coefficients of integral modular forms. Proc. Cambridge Philos. Soc., 35 : 357-372, 1939 · Zbl 0021.39202
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