Oscillations of coefficients of symmetric square L-functions over primes

Let L(s, sym(2) f) be the symmetric-square L-function associated to a primitive holomorphic cusp form f for SL(2,a"currency sign), with t (f) (n, 1) denoting the nth coefficient of the Dirichlet series for it. It is proved that, for N a (c) 3/4 2 and any alpha a a"e, there exists an effective positive constant c such that I pound (na (c) 1/2N) I >(n)t (f) (n, 1)e(n alpha) a parts per thousand(a) N , where I >(n) is the von Mangoldt function, and the implied constant only depends on f. We also study the analogue of Vinogradov's three primes theorem associated to the coefficients of Rankin-Selberg L-functions.