Oscillations of coefficients of symmetric square L-functions over primes.

Let L( s, sym f) be the symmetric-square L-function associated to a primitive holomorphic cusp form f for SL(2,ℤ), with t( n, 1) denoting the nth coefficient of the Dirichlet series for it. It is proved that, for N ⩾ 2 and any α ∈ ℝ, there exists an effective positive constant c such that ΣΛ( n) t( n, 1) e( nα) ≪ N $\exp ( - c\sqrt {\log N} )$, where Λ( 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. [ABSTRACT FROM AUTHOR]