Let \(\lambda_f(m)m^{(k-1)/2}\) denote the \(m\)th coefficient of the Fourier series at \(\infty\) of a holomorphic primitive cusp form of even weight \(k\) and the trivial character for a congruence subgroup.
In 1980’s, Rankin considered the upper and lower bounds for the moment \(S(x,\ell)= \sum_{n\leq x} |\lambda_f(n)|^{\ell}\) with \(\ell\in [0,\infty)\), \[ x(\log x)^\alpha\ll S(x,\ell)\ll x(\log x)^\beta \] for some explicit constants \(\alpha,\beta\) depending on \(\ell\) [R. A. Rankin, Math. Ann. 272, 593–600 (1985; Zbl 0556.10018)]. For the case \(\ell=1\), the upper bound was improved by G. Tenenbaum a few years ago [cf. Enseign. Math. (2) 53, No. 1–2, 155–178 (2007; Zbl 1137.11064)]. In this paper, the author sharpens many cases of Rankin’s result, and for reference, gives the conditional result \[ S(x,r)\sim C_r(f) x(\log x)^{\theta_r}\quad \text{where} \quad \theta_r = \frac{4^r\Gamma(r+1/2)}{\sqrt{\pi}\Gamma(r+2)}-1 \] and \(C_r(f)>0\) is some constant. This is now promising in light of the current vital progress (cf. [T. Barnet-Lamb et al., Publ. Res. Inst. Math. Sci. 47, No. 1, 29–98 (2011; Zbl 1264.11044)]).
In addition, the author investigated the mean value \[ S(x)=\sum_{n\leq x} \lambda_f(n). \] The result \(S(x)\ll x^{1/3+\varepsilon}\) was (essentially) known quite long time ago. In a joint work of J. L. Hafner and A. Ivić in [Enseign. Math., II. Sér. 35, No. 3–4, 375–382 (1989; Zbl 0696.10020)], one of their results was to shave off \(\varepsilon\) in the exponent. Subsequently there was a further improvement \(S(x)\ll x(\log x)^{-\gamma}\) with \(\gamma \approx 0.065\) due to a paper of R. A. Rankin in [Automorphic forms and analytic number theory, Proc. Conf., Montréal/Can. 1989, 115–121 (1990; Zbl 0735.11023)]. Now the author shows that \(\gamma = 0.118\cdots\) is admissible.