The sum of a hybrid arithmetic function over a sparse sequence

Abstract

Let $ \lambda_{f}(n) $ be the $ n $-th normalized Fourier coefficient of $ f $, which is a primitive holomorphic cusp form of even integral weight $ k\geq2 $ for the full modular group $ SL_2(\mathbb{Z}) $. Let also $ \sigma(n) $ and $ \phi(n) $ be the sum-of-divisors function and the Euler totient function, respectively. In this paper, we are able to establish the asymptotic formula of the sum of the hybrid arithmetic function $ \lambda_{f}^{l}(n)\sigma^{c}(n)\phi^{d}(n) $ over the sparse sequence $ \{n: n = a^2+b^2\} $, namely, $ \sum_{n\leq x} \lambda_{f}^{l}(n)\sigma^{c}(n)\phi^{d}(n)r_2(n) $ for $ 1\leq l\leq 8 $, where $ x $ is a sufficiently large real number, the function $ r_2(n) $ denotes the number of representations of $ n $ as $ n = a^2 + b^2 $, $ a, b, l\in \mathbb{Z} $ and $ c, d \in \mathbb{R} $.

Keywords

• fourier coefficients
• cusp forms
• automorphic $ l $-function