Shifted convolution sums of divisor functions with Fourier coefficients. (English) Zbl 1541.11012
Summary: Let \(f(z)\) be a holomorphic cusp form of weight \(\kappa\) for the full modular group \(\mathrm{SL}_2 (\mathbb{Z})\). Denote its \(n\)-th normalized Fourier coefficient by \(\lambda_f (n)\). Let \(\tau_k (n)\) denote that \(k\)-th divisor function with \(k \ge 4\). In this paper, we consider the shifted convolution sum
\[
\sum_{n \leq X} \tau_k (n) \lambda_f (n + h).
\]
We succeed in obtaining a non-trivial upper bound, which is uniform in the shift parameter \(h\).
MSC:
| 11A25 | Arithmetic functions; related numbers; inversion formulas |
| 11F30 | Fourier coefficients of automorphic forms |
| 11F11 | Holomorphic modular forms of integral weight |
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