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

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 $N_{\zeta }(\sigma ;T,T+T^a)\ll T^{15a(1-\sigma )/(3+5\sigma )+\epsilon }$ for $2/3\le a\le 1$ and $7/10\le \sigma \le 4/5$, improving Ye--Zhang's exponent in the interior of this range. Combining the localized density bound with the recent quasi-Riemann theorems, we estimate smoothed prime-power sums with phase $\exp (2\pi i\alpha n^{\beta })$ on intervals of length $K^{\delta }$. For $0<\beta <13/20$, we obtain a power saving whenever $\max \{17/30,1-\beta \}<\delta <1$; in particular the square-root phase admits every $\delta >17/30$. A revised transform estimate retains both the stationary spread and the Fourier width $K/L$. For example, the bound at $\beta =1/2$, $\delta =3/5$ is $K^{15/26+\epsilon }$, uniformly for frequencies $m\alpha$ with $1\le m\le K^{1/10}$. We also retain power savings for certain self-dual $GL(2)$ twists and self-dual $GL(3)$ cusp forms, without assuming Ramanujan. All three applications have explicit frequency dependence.