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)≪ T^{15a(1−σ)/(3+5σ)+ε} for 2/3≤ a≤1 and 7/10≤σ≤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(2πiαn^β) on intervals of length K^δ. For 0<β<13/20, we obtain a power saving whenever max{17/30,1−β}<δ<1; in particular the square-root phase admits every δ>17/30. A revised transform estimate retains both the stationary spread and the Fourier width K/L. For example, the bound at β=1/2, δ=3/5 is K^15/26+ε, uniformly for frequencies mα with 1≤ m≤ 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.

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Cite

BibTeX RIS

BibTeX

@misc{ye2026zero,
  title = {{Zero density in short strips and short-interval bounds toward Hypothesis S}},
  author = {Yangbo Ye},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.15989},
  eprint = {2610.15989},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.15989},
}

RIS

TY  - UNPB
TI  - Zero density in short strips and short-interval bounds toward Hypothesis S
AU  - Yangbo Ye
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2610.15989
UR  - https://arxiv.org/abs/2610.15989
AB  - 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)≪ T^15a(1−σ)/(3+5σ)+ε for 2/3≤ a≤1 and 7/10≤σ≤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(2πiαn^β) on intervals of length K^δ. For 0<β<13/20, we obtain a power saving whenever max{17/30,1−β}<δ<1; in particular the square-root phase admits every δ>17/30. A revised transform estimate retains both the stationary spread and the Fourier width K/L. For example, the bound at β=1/2, δ=3/5 is K^15/26+ε, uniformly for frequencies mα with 1≤ m≤ 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.
ER  -

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