## 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.