ON THE AVERAGE BEHAVIOR OF THE FOURIER COEFFICIENTS OF jTH SYMMETRIC POWER L-FUNCTION OVER CERTAIN SEQUENCES OF POSITIVE INTEGERS

We investigate the average behavior of the nth normalized Fourier coefficients of the jth ( j >= 2 be any fixed integer) symmetric power L-function (i.e., L( s, sym(j)f)), attached to a primitive holomorphic cusp form f of weight k for the full modular group SL(2, Z) over certain sequences of positive integers. Precisely, we prove an asymptotic formula with an error term for the sum [GRAPHICS] . where x is sufficiently large, and [GRAPHICS] . When j = 2, the error term which we obtain improves the earlier known result.