## On sums of Fourier coefficients of cusp forms.(English)Zbl 0696.10020

Let $$F(z)=\sum a(n)e^{2\pi inz}$$ be a cusp form of weight k for the full modular group, and set $$A(x)=\sum_{n\leq x}a(n)$$. This paper is concerned with estimates for A(x) and for the error term B(x) in the asymptotic formula $\int^{x}_{0}A(t)^ 2 dt=Cx^{k+1/2}+B(x).$ It is shown that $$A(x)=O(x^{k/2-1/6})$$ and $A(x)=\Omega_{\pm}(x^{k/2-1/4} \exp \{\frac{D(\log \log x)^{1/4}}{(\log \log \log x)^{3/4}}\}),$ for a suitable constant $$D>0$$. Moreover B(x) satisfies the same $$\Omega$$-estimate except that $$\Omega_{\pm}''$$ must be replaced by “$$\Omega$$ ”.
Reviewer: D.R.Heath-Brown

### MSC:

 11F11 Holomorphic modular forms of integral weight 11N37 Asymptotic results on arithmetic functions