A note on average behaviour of the Fourier coefficients of jth symmetric power L-function over certain sparse sequence of positive integers

Abstract

Let j ⩾ 2 be a given integer. Let H_k* be the set of all normalized primitive holomorphic cusp forms of even integral weight k ⩾ 2 for the full modulo group SL(2, ℤ). For f ∈ H_k*, denote by \({{\rm{\lambda }}_{{\rm{sy}}{{\rm{m}}^j}{\kern 1pt} f}}(n)\) the nth normalized Fourier coefficient of jth symmetric power L-function (L(s, sym^jf)) attached to f. We are interested in the average behaviour of the sum

$$\sum\limits_{\scriptstyle n\, = \,a_1^2 + a_2^2 + a_3^2 + a_4^2 + a_5^2 + a_6^2x \atop \scriptstyle \,\,\,\,\,\,\,({a_1},{a_2},{a_3},{a_4},{a_5},{a_6}{\rm{)}} \in \,{{\mathbb{Z}}^6}} {{\rm{\lambda }}_{{\rm{sy}}{{\rm{m}}^j}\,f\left( n \right),}^2}$$

where x is sufficiently large, which improves the recent work of A. Sharma and A. Sankaranarayanan (2023).