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Mean Square Estimates for Coefficients of Symmetric Power L-Functions

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Abstract

Let L(symj f,s) be the jth symmetric power L-function attached to a holomorphic Hecke eigencuspform f(z) for the full modular group, and \(\lambda_{\mathrm{sym}^{j}f}(n)\) denote its nth coefficient. In this paper we are able to prove that

$$\int_{1}^{x}\bigg|\sum_{n\leq y}\lambda_{\mathrm{sym}^{3}f}(n)\bigg|^{2}dy=O\bigl(x^{2}\bigr),$$

and

$$\int_{1}^{x}\bigg|\sum_{n\leq y}\lambda_{\mathrm{sym}^{4}f}(n)\bigg|^{2}dy=O\bigl(x^{\frac{11}{5}}\log x\bigr).$$

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  1. Department of Mathematics, Shandong Normal University, Jinan Shandong, 250014, China

    Huixue Lao

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  1. Huixue Lao
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Lao, H. Mean Square Estimates for Coefficients of Symmetric Power L-Functions. Acta Appl Math 110, 1127–1136 (2010). https://doi.org/10.1007/s10440-009-9497-2

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