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

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 $N_{\zeta }(\sigma ;T,T+T^a)\ll T^{15a(1-\sigma )\mathrm{/}(3+5\sigma )+\epsilon }$ for $2\mathrm{/}3\le a\le 1$ and $7\mathrm{/}10\le \sigma \le 4\mathrm{/}5$. This improves Ye–Zhang’s exponent for $7\mathrm{/}10<\sigma <4\mathrm{/}5$ and localizes Guth–Maynard’s density bound to short intervals of ordinates. Through the explicit formula, we apply these estimates to smoothed prime-power sums with phase $\exp (2\pi i\alpha n^{\beta })$, for fixed $\alpha \mathrm{\ne }0$ and $0<\beta <1$. We obtain explicit power savings over the interval length $K^{\delta }$ for zeta and, using the earlier cusp-form density bounds, for certain real primitive character twists of self-dual $GL(2)$ cusp forms and self-dual $GL(3)$ Maass cusp forms. The smallest interval thresholds in these three applications are $2\mathrm{/}3$, $3\mathrm{/}4$, and $5\mathrm{/}6$, respectively. We give explicit frequency dependence when $\alpha$ is replaced by $m\alpha$, uniformly for $1\le m\le K^R$ with any fixed $R>0$. The cusp-form applications do not require the Ramanujan conjecture.