# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] 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 [2] Moreno, C.J., Shahidi, F.: The fourth moment of the Ramanujan ?-function. Math. Ann.266, 233-239 (1983) · Zbl 0517.10022 [3] Ram Murty, M.: Oscillations of Fourier coefficients of modular forms. Math. Ann.262, 431-446 (1983) · Zbl 0499.10025 [4] Rankin, R.A.: An ?-result for the coefficients of cusp forms. Math. Ann.203, 239-250 (1973) · Zbl 0254.10021 [5] Rankin, R.A.: Sums of powers of cusp form coefficients. Math. Ann.263, 227-236 (1983) · Zbl 0504.10013 [6] Rankin, R.A.: A family of newforms. Ann. Acad. sci. Fenn. Ser. AI. Math.10, 93-99 (1985) · Zbl 0595.10019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.