A log-free zero-density estimate and small gaps in coefficients of L-functions

Let L(s, π × π') be the Rankin-Selberg L-function attached to automorphic representations π and π × Let π and π × denote the contragredient representations associated to π and π × Under the assumption of certain upper bounds for coefficients of the logarithmic derivatives of L(s, π × π) and L(s, π × × π × ), we prove a log-free zero-density estimate for L(s, π × π × ) which generalizes a result due to Fogels in the context of Dirichlet L-functions. We then employ this log-free estimate in studying the distribution of the Fourier coefficients of an automorphic representation π. As an application, we examine the nonlacunarity of the Fourier coefficients bf (p) of a modular newform f (z)=-∞ n=1 bf (n) e2πinz of weight k, level N, and character χ. More precisely, for f (z) and a prime p, set jf (p) :=maxx;x>p Jf (p, x), where Jf (p, x) :=#{prime q; aπ (q) =0 for all p< q ≤ x}. We prove that jf (p)-f,θ pθ for some 0<θ <1.