{"paper":{"title":"Small values of the Riemann zeta-function near the critical line","arxiv_id":"2610.27591","primary_cat":"math.NT","submitted_at":"2026-10-20T03:58:13+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":0.85,"id":"s0","own":true,"role":"contribution","text":"Assuming the Riemann Hypothesis, we study negative moments and small values of the Riemann zeta-function to the right of the critical line.","hedge":{"p":0.62,"label":"asserted"},"own_p":0.9,"main_claim_p":0.07,"quantitative":false,"quantitative_p":0.08,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.15,"limitation":0.0,"future_work":0.0,"contribution":0.85,"interpretation":0.0}},{"i":1,"p":0.99,"id":"s1","own":true,"role":"result","text":"For every fixed $0<k<1/2$, we establish the conjecturally sharp bound $$ \\frac1T\\int_T^{2T} \\Big|\\zeta(\\tfrac12+\\alpha+it)\\Big|^{-2k}\\,dt \\ll \\alpha^{-k^2}, $$ for $\\alpha=o(1)$ and $\\alpha\\gg 1/\\log T$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.96,"main_claim_p":0.53,"quantitative":true,"quantitative_p":0.98,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.99,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.0}},{"i":2,"p":0.96,"id":"s2","own":true,"role":"result","text":"We also obtain almost sharp upper bounds slightly to the left of $1/\\log T$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.93,"main_claim_p":0.08,"quantitative":false,"quantitative_p":0.44,"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":3,"p":0.58,"id":"s3","own":true,"role":"interpretation","text":"This improves the upper bounds of the first two authors for shifts as small as $1/\\log T$, confirming a conjecture of Gonek in this range.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.83,"main_claim_p":0.15,"quantitative":true,"quantitative_p":0.84,"role_probabilities":{"gap":0.01,"method":0.0,"result":0.22,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.19,"interpretation":0.58}},{"i":4,"p":0.8,"id":"s4","own":true,"role":"result","text":"A separate bound for the moment at $k=1/2$ yields an improved bound for cancellation in the M\\\"obius function.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.92,"main_claim_p":0.08,"quantitative":false,"quantitative_p":0.27,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.8,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.13,"interpretation":0.07}},{"i":5,"p":0.99,"id":"s5","own":true,"role":"result","text":"Namely, we show that $$ \\sum_{n\\leq x}\\mu(n) \\ll \\sqrt{x}\\exp \\bigg(C\\sqrt{\\frac{\\log x}{\\log\\log x}}\\bigg) $$ for some $C>0$, improving on previous results of Soundararajan, Balazard and de Roton, and of the first two authors.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.96,"main_claim_p":0.14,"quantitative":true,"quantitative_p":0.99,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.99,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.0}},{"i":6,"p":1.0,"id":"s6","own":true,"role":"result","text":"For $t$ chosen uniformly from $[T,2T]$, we also prove that \\[ \\min_{|h|\\leq1} \\Big|\\zeta\\Big(\\tfrac12+\\alpha+i(t+h)\\Big)\\Big| =\\alpha^{1+o(1)} \\] in probability, provided that $\\alpha \\ll (\\log T)^{-c}$ for some fixed $c >0$, and $\\alpha\\log T/\\log\\log T\\to\\infty$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.96,"main_claim_p":0.13,"quantitative":true,"quantitative_p":0.96,"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":7,"p":0.98,"id":"s7","own":false,"role":"interpretation","text":"This agrees with the leading order predicted by a minimum analogue of the Fyodorov--Hiary--Keating Conjecture.","hedge":{"p":0.57,"label":"asserted"},"own_p":0.45,"main_claim_p":0.08,"quantitative":false,"quantitative_p":0.42,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.02,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.98}},{"i":8,"p":0.65,"id":"s8","own":true,"role":"result","text":"Further results include bounds for minima on intervals of logarithmically growing length, for negative moments in almost all short intervals, and an extension of the range of sharp negative-moment bounds for $k>1/2$.","hedge":{"p":0.98,"label":"asserted"},"own_p":0.93,"main_claim_p":0.07,"quantitative":false,"quantitative_p":0.4,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.65,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.35,"interpretation":0.0}},{"i":9,"p":0.98,"id":"s9","own":true,"role":"method","text":"A main ingredient in the proof of some of these results is a Gaussian large-deviation upper bound for $\\log|\\zeta(1/2+\\alpha+it)|^{-1}$, obtained by building on work of Arguin, Bailey, and Roberts.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.83,"main_claim_p":0.05,"quantitative":false,"quantitative_p":0.38,"role_probabilities":{"gap":0.0,"method":0.98,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.02,"interpretation":0.0}}]},"compiled":{"main_claim":{"p":0.9,"text":"For every fixed $0<k<1/2$, we establish the conjecturally sharp bound $$ \\frac1T\\int_T^{2T} \\Big|\\zeta(\\tfrac12+\\alpha+it)\\Big|^{-2k}\\,dt \\ll \\alpha^{-k^2}, $$ for $\\alpha=o(1)$ and $\\alpha\\gg 1/\\log T$.","location":"abstract s1","hedge_strength":{"p":1.0,"label":"asserted"}},"provenance":{"route":{"via":"TypeSafe API","model":"classifier-1.13.0","endpoint":"https://api.typesafe.ai/v1/systemone","provider":"TypeSafe"},"packing":"one request per 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bound $$ \\frac1T\\int_T^{2T} \\Big|\\zeta(\\tfrac12+\\alpha+it)\\Big|^{-2k}\\,dt \\ll \\alpha^{-k^2}, $$ for $\\alpha=o(1)$ and $\\alpha\\gg 1/\\log T$.","location":"abstract s1"}},"coverage":{"tier":"abstract","levels_run":["P abstract"],"not_examined":["full text","figures and tables","references and citing sentences","relations to other papers"],"sources_read":["arXiv listing abstract v1"],"levels_not_run":["S sections","T body sentences","C citations","R cross-paper relations"],"sentences_examined":10,"sentences_in_abstract":10},"measured_accuracy":{"url":"https://pith.science/accuracy","measured_on":"2026-10-02","method":"two independent model families labelled a fresh random sample blind; the figure is how often the published label matches their 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