The average behaviour of a hybrid arithmetic function associated to cusp form coefficients over certain sparse sequence

Abstract

Let f be a normalized primitive holomorphic cusp form of even integral weight for the full modular group ? = SL(2,Z), and let ?f (n), ?(n) and ?(n) be the nth normalized Fourier coefficient of the cusp form f , the sum-of-divisors function and the Euler totient function, respectively. In this paper, we investigate the asymptotic behaviour of the following summatory function Sj,b,c(x) := X n=a21 +a22 +a23 +a24 ?x (a1,a2,a3,a4)?Z4 ?j f (n)?b(n)?c(n), where j ? 2 is any given integer. In a similar manner, we also establish other similar results related to normalized coefficients of the symmetric power L-functions associated to holomorphic cusp form f.