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Sums of powers of cusp form coefficients. II. (English) Zbl 0556.10018
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 C. J. Moreno and 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 P. D. T. A. Elliott, C. J. Moreno and 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).

##### MSC:
 11F11 Holomorphic modular forms of integral weight 11N37 Asymptotic results on arithmetic functions
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##### References:
  Elliott, P.D.T.A., Moreno, C.J., Shahidi, F.: On the absolute value of Ramanujan’s ?-function. Math. Ann.266, 507-511 (1984) · Zbl 0531.10030  Moreno, C.J., Shahidi, F.: The fourth moment of the Ramanujan ?-function. Math. Ann.266, 233-239 (1983) · Zbl 0517.10022  Ram Murty, M.: Oscillations of Fourier coefficients of modular forms. Math. Ann.262, 431-446 (1983) · Zbl 0499.10025  Rankin, R.A.: An ?-result for the coefficients of cusp forms. Math. Ann.203, 239-250 (1973) · Zbl 0254.10021  Rankin, R.A.: Sums of powers of cusp form coefficients. Math. Ann.263, 227-236 (1983) · Zbl 0504.10013  Rankin, R.A.: A family of newforms. Ann. Acad. sci. Fenn. Ser. AI. Math.10, 93-99 (1985) · Zbl 0595.10019
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