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


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