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Keywords:
Hecke eigenform; Fourier coefficient; Rankin-Selberg $L$-function
Hecke eigenform; Fourier coefficient; Rankin-Selberg $L$-function
Summary:
Let $f$, $g$ and $h$ be three distinct primitive holomorphic cusp forms of even integral weights $k_{1}$, $k_{2}$ and $k_{3}$ for the full modular group $\Gamma ={\rm SL}(2,\mathbb {Z})$, respectively, and let $\lambda _{f}(n)$, $\lambda _{g}(n)$ and $\lambda _{h}(n)$ denote the $n$th normalized Fourier coefficients of $f$, $g$ and $h$, respectively. We consider the cancellations of sums related to arithmetic functions $\lambda _{g}(n)$, $\lambda _{h}(n)$ twisted by $\lambda _{f}(n)$ and establish the following results: $$ \sum _{n\leq x}\lambda _{f}(n)\lambda _{g}(n)^{i}\lambda _{h}(n)^{j} \ll _{f,g,h,\varepsilon } x^{1- 1/2^{i+j} +\varepsilon } $$ for any $\varepsilon >0$, where $1\leq i\leq 2$, $j\geq 5$ are any fixed positive integers.
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