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 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 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):=max x; x>p J f (p,x) , where J f (p,x):=#{prime q; a π (q)=0 for all p<q≤x}. We prove that j f (p)≪ f,θ p θ for some 0<θ<1 . Show more