Abstract
Let F(Z) be a cusp form of integral weight k relative to the Siegel modular group Spn(Z) and let f(N) be its Fourier coefficient with index N. Making use of Rankin's convolution, one proves the estimate
where
Previously, for n ≥ 2 one has known Raghavan's estimate
In the case n=2, Kitaoka has obtained a result, sharper than (1), namely:
At the end of the paper one investigates specially the case n=2. It is shown that in some cases the result (2) can be improved to, apparently, unimprovable estimates if one assumes some analogues of the Petersson conjecture. These results lead to a conjecture regarding the optimal estimates of f(N), n=2.
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Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 144, pp. 155–166, 1985.
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Fomenko, O.M. Fourier coefficients of Siegel cusp forms of genus n. J Math Sci 38, 2148–2157 (1987). https://doi.org/10.1007/BF01474450
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DOI: https://doi.org/10.1007/BF01474450