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Shifted convolution sums of fourier coefficients with divisor functions

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Abstract

Let f(z) be a primitive holomorphic cusp form of even integral weight k for the full modular group. Denote its nth normalized Fourier coefficient (Hecke eigenvalue) by λ f (n). Let d(n) be the Dirichlet divisor function. In this paper, we establish that

$$\left.\begin{array}{ll}\quad \sum\limits_{n\leqq x}|\lambda_{f}(n^j)|d(n - 1) = c(f, j)x(log x)^{{\delta}_{j}} (1 + o(1)),\\\sum\limits_{n\leqq x}|\lambda_{f}(n^i)\lambda_{f}(n^j)|d(n - 1) = c(f, i, j)x(log x)^{{\delta}_{i,j}} (1 + o(1)),\\\quad \sum\limits_{n\leqq x}\lambda_{f}(n)^{2j}d(n - 1) = d(f, j)x(log x)^{{A}_{j}} (1 + o(1)),\end{array}\right.$$

where \({0 < \delta_{j} < 1}\), \({0 < \delta_{i,j}\leqq 1}\) (\({\delta_{i,j} = 1}\) if and only if i = j), and

$$A_j = (2j)!/(j!(j + 1)!).$$

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  1. School of Mathematics, Shandong University, Jinan Shandong, 250100, China

    G. Lü

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Lü, G. Shifted convolution sums of fourier coefficients with divisor functions. Acta Math. Hungar. 146, 86–97 (2015). https://doi.org/10.1007/s10474-015-0499-4

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