Sums of powers of cusp form coefficients. II (Q761485)

Let f be a newform of weight k for the modular group SL(2,\({\mathbb{Z}})\) given by \(f(z)=\sum^{\infty}_{n=1}a(n)e^{2\pi inz}\) and write \(\alpha (n)=a(n)n^{(1-k)/2}.\) The author continues his study, begun in [ibid. 263, 227-236 (1983; Zbl 0492.10020)] of the sums \(S(x,2\beta)=\sum_{n\leq x}| \alpha (n)|^{2\beta}\) for large x and \(\beta\geq 0\). By using the result of \textit{C. J. Moreno} and \textit{F. Shahidi} [ibid. 266, 233-239 (1983; Zbl 0508.10014)] that \(S(x,4)\sim A_ 2 x \log x,\) in addition to the classical result \(S(x,2)\sim A_ 1 x\), he obtains improved upper and lower bounds for all \(\beta\geq 0\). Write \(B_ 1=[0,1]\cup [2,\infty]\) and \(B_ 2=[1,2]\); then it is shown that \[ x(\log x)^{G(\beta)}\ll S(x,2\beta)\ll x(\log x)^{F(\beta)}\quad for\quad x\in B_ 1, \] \[ x(\log x)^{F(\beta)}\ll S(x,2\beta)\ll x(\log x)^{G(\beta)}\quad for\quad x\in B_ 2, \] where \(F(\beta)=(1/5)2^{\beta -1}(2^{\beta}+3^{2- \beta})-1\) and \(G(\beta)=2^{\beta -1}-1\). In particular, for \(\beta =\), \[ x(\log x)^{-G}\ll \sum_{n\leq x}| \alpha (n)| \ll x(\log x)^{-F}, \] where \(F=(8+3\sqrt{6})^{-1}=0.0652\) and \(G=1-2^{- 1/2}=0.2962\). In the previously obtained best resuls the exponents F and G were replaced, respectively, by 1/18 [obtained by \textit{P. D. T. A. Elliott, C. J. Moreno} and \textit{F. Shahidi}, Math. Ann. 266, 507-511 (1984; Zbl 0513.10024)] and the author (op. cit.). Similar results are obtained for the sums \(\sum_{p\leq x}| \alpha (p)|^{2\beta}\) (p a prime).