{"paper":{"title":"Zero density in short strips and short-interval bounds toward Hypothesis S","arxiv_id":"2610.15989","primary_cat":"math.NT","submitted_at":"2026-10-11T23:39:47+00:00","version_compiled":"v1"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13.0 on the question bank abstract-p1-20260923, exactly as returned. No sentence in this record was written by a language model.","schema":"pith.compiled-record/v0","status":"compiled","abstract":{"source":"arXiv listing abstract v1","sentences":[{"i":0,"p":0.59,"id":"s0","own":true,"role":"contribution","text":"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.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.94,"main_claim_p":0.33,"quantitative":false,"quantitative_p":0.06,"role_probabilities":{"gap":0.0,"method":0.4,"result":0.01,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.59,"interpretation":0.0}},{"i":1,"p":1.0,"id":"s1","own":true,"role":"result","text":"For the Riemann zeta function, we obtain $N_\\zeta(\\sigma;T,T+T^a)\\ll T^{15a(1-\\sigma)/(3+5\\sigma)+\\varepsilon}$ for $2/3\\le a\\le1$ and $7/10\\le\\sigma\\le4/5$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.94,"main_claim_p":0.38,"quantitative":true,"quantitative_p":0.99,"role_probabilities":{"gap":0.0,"method":0.0,"result":1.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":2,"p":0.86,"id":"s2","own":true,"role":"interpretation","text":"This improves Ye--Zhang's exponent for $7/10<\\sigma<4/5$ and localizes Guth--Maynard's density bound to short intervals of ordinates.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.9,"main_claim_p":0.3,"quantitative":true,"quantitative_p":0.9,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.08,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.06,"interpretation":0.86}},{"i":3,"p":0.72,"id":"s3","own":true,"role":"method","text":"Through the explicit formula, we apply these estimates to smoothed prime-power sums with phase $\\exp(2\\pi i\\alpha n^\\beta)$, for fixed $\\alpha\\ne0$ and $0<\\beta<1$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.92,"main_claim_p":0.06,"quantitative":false,"quantitative_p":0.38,"role_probabilities":{"gap":0.0,"method":0.72,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.28,"interpretation":0.0}},{"i":4,"p":0.96,"id":"s4","own":true,"role":"result","text":"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.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.95,"main_claim_p":0.19,"quantitative":true,"quantitative_p":0.7,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.96,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.04,"interpretation":0.0}},{"i":5,"p":1.0,"id":"s5","own":true,"role":"result","text":"The smallest interval thresholds in these three applications are $2/3$, $3/4$, and $5/6$, respectively.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.9,"main_claim_p":0.11,"quantitative":true,"quantitative_p":0.98,"role_probabilities":{"gap":0.0,"method":0.0,"result":1.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":6,"p":0.72,"id":"s6","own":true,"role":"result","text":"We give explicit frequency dependence when $\\alpha$ is replaced by $m\\alpha$, uniformly for $1\\le m\\le K^R$ with any fixed $R>0$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.93,"main_claim_p":0.04,"quantitative":true,"quantitative_p":0.78,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.72,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.28,"interpretation":0.0}},{"i":7,"p":0.68,"id":"s7","own":true,"role":"result","text":"The cusp-form applications do not require the Ramanujan 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