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

Let $L(s,\pi \times {\pi }^{\mathrm{\prime }})$ be the Rankin–Selberg $L$-function attached to automorphic representations $\pi$ and ${\pi }^{\mathrm{\prime }}$. Let $\tilde{\pi }$ and ${\tilde{\pi }}^{\mathrm{\prime }}$ denote the contragredient representations associated to $\pi$ and ${\pi }^{\mathrm{\prime }}$. Under the assumption of certain upper bounds for coefficients of the logarithmic derivatives of $L(s,\pi \times \tilde{\pi })$ and $L(s,{\pi }^{\mathrm{\prime }}\times {\tilde{\pi }}^{\mathrm{\prime }})$, we prove a log-free zero-density estimate for $L(s,\pi \times {\pi }^{\mathrm{\prime }})$ 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 $f(z)={\sum }_{n=1}^{\mathrm{\infty }}{b}_f(n)e^{2\pi inz}$ 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 $J_{f} (p, x):=#{{\rm prime}~q;~a_{\pi}(q)=0~{\rm for~all~}p<q\leq x}.$Weprovethat$j_f(p)\ll_{f, \theta} p^\theta$forsome$0<\theta<1$.