# zbMATH — the first resource for mathematics

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
Full Text:
##### 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
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.