Oscillations of coefficients of symmetric square L-functions over primes

Abstract

Let L(s, sym²f) be the symmetric-square L-function associated to a primitive holomorphic cusp form f for SL(2,\mathbf{\mathbb{Z}}), with t_f(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 {\displaystyle {\sum }_{n\le N}\mathrm{\Lambda }(n)t_f(n,\mathrm{1})e(n\alpha )\ll 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.

Key words： symmetric-square L-function; primitive holomorphic cusp form; Fourier coefficient

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