Zero density estimates for automorphic \(L\)-functions of \(\mathrm{GL}_m\) (Q343301)

This paper is concerned with zero-density estimates for automorphic \(L\)-functions \(L(s,\pi)\) for \(\mathrm{GL}_m\). These are estimates on the number \(N_\pi(\sigma,S,T)\) of zeros \(\rho=\beta+i\gamma\) of \(L(s,\pi)\) with \(\sigma>\beta<1\) and \(S\leq \gamma\leq T\). Typically, such estimates come conditionally on some moment growth condition, for example, it is shown that if \[ \int_T^{T+T^\alpha}\left| L\left(\tfrac12+it,\pi\right)\right| ^{2l}\,dt \ll_{\varepsilon,\pi}T^{\theta+\varepsilon} \] for some \(0<\alpha\leq 1\) and \(\theta\geq\alpha\), then \[ N_\pi(\sigma,T,T+T^\alpha)\ll T^{2(1-\sigma)+\varepsilon} \] for \(\frac12\leq \sigma <1\). A number of results in this direction is given. The proof use the Halász-Montgomery inequality and bounds for moments of \(L\).