Discrete mean square estimates for coefficients of symmetric power -functions
Tom 190 / 2019
Acta Arithmetica 190 (2019), 193-208
MSC: 11F30, 11F66.
DOI: 10.4064/aa180819-6-10
Opublikowany online: 27 June 2019
Streszczenie
Let f be a primitive holomorphic Hecke eigenform for \mathrm{SL}(2, \mathbb{Z}). Let L(\mathop{\rm sym}\nolimits^j f, s) be the jth symmetric power L-function associated to f, and \lambda_{\mathop{\rm sym}\nolimits^j f}(n) its nth Fourier coefficient. We prove asymptotic formulas for the sums \begin{equation*} \sum_{n \leq x} | \lambda_{\mathop{\rm sym}\nolimits^3 f}(n)|^2 \quad \text{and} \quad \sum_{n \leq x} | \lambda_{\mathop{\rm sym}\nolimits^4 f}(n)|^2 \end{equation*} with improved error terms for x\geq x_0 (large).
