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 $N_ζ(σ;T,T+T^a)\ll T^{15a(1-σ)/(3+5σ)+\varepsilon}$ for $2/3\le a\le1$ and $7/10\leσ\le4/5$. This improves Ye--Zhang's exponent for $7/10<σ<4/5$ 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 $\exp(2πiαn^β)$, for fixed $α\ne0$ and $0<β<1$. We obtain explicit power savings over the interval length $K^δ$ 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 $2/3$, $3/4$, and $5/6$, respectively. We give explicit frequency dependence when $α$ is replaced by $mα$, uniformly for $1\le m\le K^R$ with any fixed $R>0$. The cusp-form applications do not require the Ramanujan conjecture.