A Log-free Zero-density Estimate and Small Gaps in Coefficients of L-functions

Abstract. Let L(s,π × π) be the Rankin–Selberg L-function attached to automorphic representations π and π. Let π̃ and π̃ 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 × π̃) and L(s,π × π̃), we prove a log-free zero-density estimate for L(s,π × π) which generalises 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 non-lacunarity of the Fourier coefficients b f (p) of a modular newform f (z) = ∑∞ n=1 b f (n)e 2πinz of weight k, level N, and character χ. More precisely for f (z) and a prime p, set j f (p) := maxx; x>p J f (p, x), where J f (p, x) := #{prime q; aπ(q) = 0 for all p < q ≤ x}. We prove that j f (p) ≪ f ,θ p for some 0 < θ < 1.