Zero density in short strips and short-interval bounds toward Hypothesis S

Source abstract

We deduce zero-density estimates in short vertical strips from short-interval moments of globally tempered automorphic $L$-functions, by adapting the large-value estimate of Guth and Maynard and the Type II zero argument of Maynard and Pratt to the framework of Ye and Zhang. For the Riemann zeta function, we obtain $N_ζ(σ;T,T+T^a)\ll T^{15a(1-σ)/(3+5σ)+\varepsilon}$ for $2/3\le a\le1$ and $7/10\leσ\le4/5$. This improves Ye--Zhang's exponent for $7/10 0$. The cusp-form applications do not require the Ramanujan conjecture.