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

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 $f(z)=\sum_{n=1}^{\infty} b_f(n) e^{{2\pi i n z}}$ 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 $J_{f} (p, x):=\#\{{\rm prime}~q;~a_{\pi}(q)=0~{\rm for~all~}p<q\leq x\}.$ We prove that $j_f(p)\ll_{f, \theta} p^\theta$ for some $0<\theta<1$.