Sums of cusp form coefficients. (English) Zbl 0735.11023

Automorphic forms and analytic number theory, Proc. Conf., Montréal/Can. 1989, 115-121 (1990).
[For the entire collection see Zbl 0728.00008.]
Let \(A^*(x)=\sum_{n\leq x}a^*(n)\), \(a^*(n)=n^{(1-k)/2}a(n)\), where \(a(n)\) is the \(n\)-th Fourier coefficient of a cusp form \(F\) of integral weight \(k\) belonging to the group \(\Gamma(1)\) or, possibly, to one of its congruence subgroups. The author proves that, for any \(\epsilon>0\), \[ A^*(x)\ll_ \epsilon x^{1/3}(\log x)^{- \delta+\epsilon}, \delta={8-3\sqrt 6\over 10}=0.065153\dots.(1) \] This result improves the bound \(A^*(x)\ll x^{1/3}\) obtained by J. L. Hafner and the reviewer [Enseign. Math., II. Sér. 35, 375-382 (1989; Zbl 0696.10020)], who also proved \[ A^*(x)=\Omega_ \pm (x^{1/4} \exp ({D(\log\log x)^{1/4}\over (\log\log\log x)^{3/4}}))\quad\quad (D>0).(2) \] From classical estimates of analytic number theory it is seen that the estimation of \(A^*(x)\) may be reduced to the estimation of \(\sum_{x<n\leq x+z}| a^*(n)|\)   \((x^ \epsilon<z\leq x)\). The author shows then how good bounds for the last sum may be obtained by appealing to his result on sums of \(| a^*(p)|^{2\beta}\) (\(p\) prime) [Math. Ann. 263, 227-236 (1983; Zbl 0492.10020) and ibid. 272, 593-600 (1985; Zbl 0556.10018)]. The bound (1) is interesting because of the negative exponent of the logarithm. It is conjectured that \(A^*(x)\ll_ \epsilon x^{{1\over 4}+\epsilon}\) (this is supported by (2)), and improvements on the exponent 1/3 of \(x\) in (1) would result from a nontrivial estimation of the sum \(\sum_{N<n\leq 2N}a(n)e(\sqrt{nx})\). This, however, appears to be a problem of exceptional difficulty.
Reviewer: A.Ivić (Beograd)


11F30 Fourier coefficients of automorphic forms
11F12 Automorphic forms, one variable
11N37 Asymptotic results on arithmetic functions
11L05 Gauss and Kloosterman sums; generalizations