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

Abstract

Let $L(s,\pi \times {\pi }^{\prime })$ be the Rankin--Selberg $L$-function attached to automorphic representations $\pi$ and ${\pi }^{\prime }$. Let $\tilde{\pi }$ and ${\tilde{\pi }}^{\prime }$ denote the contragredient representations associated to $\pi$ and ${\pi }^{\prime }$. Under the assumption of certain upper bounds for coefficients of the logarithmic derivatives of $L(s,\pi \times \tilde{\pi })$ and $L(s,{\pi }^{\prime }\times {\tilde{\pi }}^{\prime })$, we prove a log-free zero-density estimate for $L(s,\pi \times {\pi }^{\prime })$ 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 $\pi$. As an application we examine the non-lacunarity of the Fourier coefficients $b_f(p)$ of a modular newform $f(z)={\sum }_{n=1}^{\infty }{b}_f(n)e^{2\pi inz}$ of weight $k$, level $N$, and character $\chi$. More precisely for $f(z)$ and a prime $p$, set $j_f(p):={\max }_{x;x>p}{J}_f(p,x)$, where $J_f(p,x):=\#\{\mathrm{prime}q;a_{\pi }(q)=0\mathrm{for}\mathrm{all}p<q\le x\}.$ We prove that $j_f(p){\ll }_{f,\theta }{p}^{\theta }$ for some $0<\theta <1$.