Fourier coefficients of symmetric power \(L\)-functions. (English) Zbl 1417.11050
Summary: Let \(f\) be a Hecke eigencusp form of even integral weight \(k\) or Maass cusp form for the full modular group \(SL_2(\mathbb{Z})\). Denote by \(\lambda_{\operatorname{sym}^m f}(n)\) the \(n\)th normalized coefficient of the Dirichlet expansion of the \(m\)th symmetric power \(L\)-function associated to \(f\). In this paper, we establish some bounds for
\[
\mathop{\sum}\limits_{n \leqslant x} \lambda_{\operatorname{sym}^m f}(n), \mathop{\sum}\limits_{n \leqslant x} \lambda_f(n^m),
\]
which improve the corresponding results of Y.-K. Lau and G. Lü [Q. J. Math. 62, No. 3, 687–716 (2011; Zbl 1269.11044)].
MSC:
| 11F30 | Fourier coefficients of automorphic forms |
| 11F11 | Holomorphic modular forms of integral weight |
| 11F66 | Langlands \(L\)-functions; one variable Dirichlet series and functional equations |
Citations:
Zbl 1269.11044References:
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